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PH507 Multi-wavelength Professor Michael Smith THE MULTIWAVELENGTH UNIVERSE AND EXOPLANETS School of Physical Sciences Convenor Prof. Michael Smith Taught in Term 2 Teaching Provision: 1 PH507 ECTS Credits 7.5 Kent Credits 15 at Level I 30 lectures + 4 workshops + 2 class tests Prerequisites: PH300, PH301, PH304 Aims: To provide a basic but rigorous grounding in observational, computational and theoretical aspects of astrophysics to build on the descriptive course in Part I, and to consider evidence for the existence of exoplanets in other Solar Systems. Learning Outcomes: 1. An understanding of the fundamentals of making astronomical observations across the whole electromagnetic spectrum, including discussion of photometry and spectroscopy, and the physics of the astrophysical radiation mechanisms. 2. An understanding of the motions of objects in extrasolar systems and the basic techniques required to solve the 2-body problem to measure their properties. 3. An understanding of observational characteristics of stars, and how their physical structures are derived from observation and using simple physical models. 4. To be able to discuss coherently the origin and evolution of Solar Systems and be able to evaluate claims for evidence of Solar Systems other than our own. SYLLABUS: • • • • • • Part 1: measurements Part 2: radiation Part 3: dynamics Part 4: star and planet formation Part 5: telescopes/instruments Part 6: stars and stellar structure Assessment Methods: Examination 70%, Homework 10%, 1st class test 10%, 2nd class test 10%. Recommended Texts: Carroll & Ostlie, An Introduction to Modern Astrophysics, Addison-Wesley, [QB461] Stuart Clark. Extrasolar Planets, Wiley Press C.R.Kitchin. Astrophysical Techniques, Adam Hilger Press. [Note: Changes may occur to the syllabus during the year] Prof. Michael Smith: 101 Ingram, x7654, [email protected] Office hours: 10-12am Wed Bad weather Numbers, names Locations, times of lectures Lecturers PH507 Multi-wavelength Professor Michael Smith 2 PART 1: Measurement LECTURE 1 Distance: Distance is an easy concept to understand: it is just a length in some units such as in feet, km, light years, parsecs etc. It has been excrutiatingly difficult to measure astronomical distances until this century. Unfortunately most stars are so far away that it is impossible to directly measure the distance using the classic technique of triangulation. Trignometric parallax: based on triangulation – need three parameters to fully define any triangle e.g. two angles and one baseline. To triangulate to even the closest stars we would need to use a very large baseline. In fact we do have a long baseline, because every 6 months the earth is on opposite sides of the sun. So we can use as a baseline the major axis of the earth's orbit around the sun. BASELINE: 2 x earth-sun distance = 2 Astronomical Units (AU) (The average distance from the earth to the sun is called the Astronomical Unit.) The parallactic displacement of a star on the sky as a result of the Earth’s orbital motion permits us to determine the distance from the Sun to the star by the method of trigonometric (heliocentric) parallax. We define the trigonometric parallax of the star as the angle subtended, as seen from the star, by the Earth’s orbit of radius 1 AU. If the star is at rest with respect to the Sun, the parallax is half the maximum apparent annual angular displacement of the star as seen from the Earth. PH507 Multi-wavelength Professor Michael Smith 3 PH507 Multi-wavelength Professor Michael Smith 4 PH507 Multi-wavelength Professor Michael Smith 5 1 radian is defined as: 360 57.3 degrees = 206265 arc seconds, approximately. There are 2 2 rad in a circle (360˚), so that 1 radian equals 57˚17´44.81” (206, 264.81”). 1 radian = Independent distance unit is the light year: c t ( year ) 9.47 1015 m The light year is not used much by professional astronomers, who work instead with a unit of similar size called the parsec, where 1 parsec = 1 pc = 206265 AU = 3.086 x 1016 m = 3.26 light years. The measurement and interpretation of stellar parallaxes are a branch of astrometry, and the work is exacting and time-consuming. Consider that the nearest star, Proxima/Alpha Centauri (Rigil Kent), at a distance of 1.3 pc, has a parallax of only 0.76”; all other stars have smaller parallaxes. PH507 Multi-wavelength Professor Michael Smith 6 Formula: tan p 1AU d or d 1 AU p where p is in radians for small angles. To convert to arcseconds: 2.063 105 d AU p '' or d 1 pc . p" Technological advances (including the Hubble Space Telescope) have improved parallax accuracy to 0.001” within a few years. Before 1990, fewer than 10,000 stellar parallaxes had been measured (and only 500 known well), but there are about 1012 stars in our Galaxy. Space observations made by the European Space Agency with the Hipparcos mission (1989-1993) accurately determined the parallaxes of many more stars. Though a poor orbit limited its usefulness, Hipparcos was expected to achieve a precision of about 0.002”. It actually achieved 0.001” for 118,000 stars. The method of trigonometric parallax is important because it is our only direct distance technique for stars. The ground-based trigonometric parallax of a star is determined by photographing a given star field from a number (about 20) of selected points in the Earth’s orbit. The comparison stars selected are distant background stars of nearly the same apparent brightness as the star whose parallax is being measured. Corrections are made for atmospheric refraction and dispersion and for detectable motions of the background stars; any motion of the star relative to the Sun is then extracted. What remains is the smaller annual parallactic motion; it is recognised because it cycles annually. Because a seeing resolution of 0.25” is considered exceptional (more typical it is 1”), it may seem strange that a stellar position can be determined to ±0.01” in one measurement; this accuracy is possible because we are determining the centre of the fuzzy stellar image. PH507 Multi-wavelength Professor Michael Smith 7 In 2011 – 2013, Gaia will be set into orbit with a Soyuz rocket (and SIM Space Interferometric Mission from the US). It will be able to measure parallaxes of 10 micro-arcseconds. It consists of a rotating frame holding three telescopes. Some aims: …….Accurate distances even to the Galactic centre, 8000 parsecs away. ……..Photometry: accurate magnitudes. ……..Planet quest ……..Reference frame from distant quasars (3C273 is 800 Mpc away) In the meantime, to go further, we construct the COSMIC LADDER. If we can estimate the luminosity of a star from other properties, they can be used as STANDARD CANDLES. 2 LUMINOSITY. We can actually only measure the radiant flux of a flame and need to make a few assumptions to find the true luminosity. Luminosity depends on the distance and extinction (as well as relativistic effects). The measured flux f is in units of W/m2 , the flow of energy per unit area. The radiated power L, ignoring extinction, is given by: f d2 L 4d 2 L 4f ’ showing that a standard candle can yield the distance. The Stellar Magnitude Scale The first stellar brightness scale - the magnitude scale - was defined by Hipparchus of Nicea and refined by Ptolemy almost 2000 years ago. In this qualitative scheme, naked-eye stars fall into six categories: the brightest are of first magnitude, and the faintest of sixth magnitude. Note that the brighter the star, the smaller the value of the magnitude. In 1856, N. R. Pogson verified William Herschel’s finding that a first-magnitude star is 100 times brighter than a sixthmagnitude star and the scale was quantified. Because an interval of five magnitudes corresponds to a factor of 100 in brightness, a one-magnitude difference corresponds to a factor of 1001/5 = 2.512. (This definition reflects the PH507 Multi-wavelength Professor Michael Smith 8 operation of human vision, which converts equal ratios of actual intensity to equal intervals of perceived intensity. In other words, the eye is a logarithmic detector). The magnitude scale has been extended to positive magnitudes larger than +6.0 to include faint stars (the 5-m telescope on Mount Palomar can reach to magnitude +23.5) and to negative magnitudes for very bright objects (the star Sirius is magnitude -1.4). The limiting magnitude of the Hubble Space Telescope is about +30. Astronomers find it convenient to work with logarithms to base 10 rather than with exponents in making the conversions from brightness ratios to magnitudes and vice versa. Consider two stars of magnitude m and n with respective apparent brightnesses (fluxes) lm and ln. The ratio of their fluxes fn / fm corresponds to the magnitude difference m - n. Because a one-magnitude difference means a brightness ratio of 1001/5, (m - n) magnitudes refer to a ratio of (1001/5)m-n = 100(m-n)/5, or fn / fm = 100(m-n)/5 Taking the log10 of both sides (because log xa = a log x and log 10a = a log 10 = a), log (fn / fm) = [(m - n)/5] log 100 = 0.4(m - n) or m - n = 2.5 log (fn / fm) This last equation defines the apparent magnitude; note that m > n when fn > fm, that is: brighter objects have numerically smaller magnitudes. Also note that when the brightnesses are those observed at the Earth, physically they are fluxes. Apparent magnitude is the astronomically peculiar way of talking about fluxes. Here are a few worked examples: PH507 Multi-wavelength Professor Michael Smith 9 (a) The apparent magnitude of the variable star RR Lyrae ranges from 7.1 to 7.8 - a magnitude amplitude of 0.7. To find the relative increase in brightness from mini-mum to maximum, we use log (fmax / fmin) = 0.4 x 0.7 = 0.28 so that fmax / fmin = 100.28 = 1.91 This star is almost twice as bright at maximum light than at minimum. (b) A binary system consists of two stars a and b, with a brightness ratio of 2; however, we see them unresolved as a point of magnitude +5.0. We would like to find the magnitude of each star. The magnitude difference is mb - ma = 2.5 log (fa / fb) = 2.5 log 2 = 0.75 Since we are dealing with brightness ratios, it is not right to put ma + mb = +5.0. The sum of the luminosities (fa + fb) corresponds to a fifth-magnitude star. Compare this to a 100-fold brighter star, of magnitude 0.0 and luminosity l0: PH507 Multi-wavelength Professor Michael Smith 10 ma+ b - m0 = 2.5 log [l0 / (fa + fb)] or 5.0 - 0.0 = 2.5 log 100 = 5. But fa = 2 fb, so that fb = (fa + fb)/3. Therefore (mb - m0) = 2.5 log (f0 / fb) = 2.5 log 300 = 2.5 x 2.477 = 6.19. The magnitude of the fainter star is 6.19, and from our earlier result on the magnitude difference, that of the brighter star is 5.44. What units are used in astronomical photometry? The well-known magnitude scale of course, which has been calibrated using standard stars whic vary in brightness. But how does the astronomical magnitude scale relate to other photometric units? Here we assum unless otherwise noted, which are at least approximately convertible to lumes, candelas, and lux 1 mv=0 star outside Earth's atmosphere = 2.54 10-6 lux = 2.54 10-10 phot Luminance: ( 1 nit =1 candela per square metre) 1 mv=0 star per sq degree outside Earth's atmosphere = 0.84E-2 nit = 8.4 10-7 stilb 1 mv=0 star per sq degree inside clear unit airmass = 6.9 10-7 stilb = 0.69E-2 nit (1 clear unit airmass transmits 82% in the visual, i.e. it dims 0.2 magnitudes) One star, Mv=0 outside Earth's atmosphere = 2.451029 cd Apparent magnitude is thus an irradiance or illuminance, i.e. incident flux per unit area, from all directions. Of course a star is a point light source, and the incident light is only from one direction. Apparent magnitude per square degree is a radiance, luminance, intensity, or "specific intensity". This is sometimes also called "surface brightness". Still another unit for intensity is magnitudes per square arcsec, which is the magnitude at which each square arcsec of an extended light source shines. Only visual magnitudes can be converted to photometric units. U, B, R or I magnitudes are not easily convertible to luxes, lumens and friends, because of the different wavelengths intervals used. The conversion factors would be strongly dependent on e.g. the temperature of the blackbody PH507 Multi-wavelength Professor Michael Smith 11 radiation or, more generally, the spectral distribution of the radiation. The conversion factors between V magnitudes and photometric units are only slightly dependent on the spectral distribution of the radiation. What units are used in radiometry/infrared astronomy? Here we're not interested in the photometric response of some detector with a well-known passband (e.g. the human eye, or some astronomical photometer). Instead we want to know the strength of the radiation in absolute units: watts etc. Thus we have: Radiance, intensity or specific intensity: W m-2 ster-1 [Å-1] SI unit erg cm-2 s-1 ster-1 [Å-1] CGS unit photons cm-2 s-1 ster-1 [Å-1] Photon flux, CGS units Irradiance/emittance, or flux: W m-2 [Å-1] SI unit -1 erg cm-2 s-1 [Å ] CGS unit photons cm-2 s-1 ster-1 [Å-1] Photon flux, CGS units Note the [A-1] within brackets. Fluxes and intensities can be total (summed over all wavelengths) or monochromatic ("per Angstrom Å" or "per nanometer"). In Radio/Infrared Astronomy, the unit Jansky is often used as a measure of irradiance at a specific wavelength, and is the radio astronomer's equivalence to stellar magnitudes. The Jansky is defined as: 1 Jansky = 10-26 W m-2 Hz-1 Absolute magnitude represents a total flux, expressed in e.g. candela, or lumens. Absolute Magnitude and Distance Modulus So far we have dealt with stars as we see them, that is, their fluxes or apparent magnitudes, but we want to know the luminosity of a star. A very luminous star will appear dim if it is far enough away, and a low-luminosity star may look bright if it is close enough. Our Sun is a case in point: if it were at the distance of the closest star (Alpha Centauri), the Sun would appear slightly fainter to us than Alpha Centauri does. Hence, distance links fluxes and luminosities. PH507 Multi-wavelength Professor Michael Smith 12 The luminosity of a star relates to its absolute magnitude, which is the magnitude that would be observed if the star were placed at a distance of 10 pc from the Sun. (Note that absolute magnitude is the way of talking about luminosity peculiar to astronomy). By convention, absolute magnitude is capitalised (M) and apparent magnitude is written lowercase (m). The inversesquare law of radiative flux links the flux f of a star at a distance d to the luminosity F it would have it if were at a distance D = 10 pc: F / f = (d / D)2 = (d / 10) 2. If M corresponds to L and m corresponds to luminosity l, then m - M = 2.5 log (F / f ) = 2.5 log (d/10)2 = 5 log (d / 10) Expanding this expression, we have useful alternative forms: since m1 m2 2.5 log d1 5 log d1 5 log d2 , d2 defining the absolute magnitude m2 = M at d2 = 10 pc, so m1 = m and d2 = d, m - M = 5 log d - 5 M = m + 5 - 5 log d In terms of the parallax, M = m + 5 + 5 log p” Here d is in parsecs and p” is the parallax angle in arc seconds. PH507 Multi-wavelength Professor Michael Smith 13 The quantity m - M is called the distance modulus, for it is directly related to the star’s distance. In many applications, we refer only to the distance moduli of different objects rather than converting back to distances in parsecs or lightyears. Magnitudes at Different Wavelengths The kind of magnitude that we measure depends on how the light is filtered anywhere along the path of the detector and on the response function of the detector itself. So that problem comes down to how to define standard magnitude systems. PH507 Multi-wavelength Professor Michael Smith 14 Magnitude Systems Detectors of electromagnetic radiation (such as the photographic plate, the photoelectric photometer, and the human eye) are sensitive only over given wavelength bands. So a given measurement samples but part of the radiation arriving from a star. Four images of the Sun, made using (a) visible light, (b) ultraviolet light, (c) X rays, and (d) radio waves. By studying the similarities and differences among these views of the same object, important clues to its structure and composition can be found. Because the flux of starlight varies with wavelength, the magnitude of a star depends upon the wavelength at which we observe. Originally, photographic plates were sensitive only to blue light, and the term photographic magnitude (mpg) still refers to magnitudes centred around 420 nm (in the blue region of the spectrum). Similarly, because the human eye is most sensitive to green and yellow, visual magnitude (mv) or the photographic equivalent photo visual magnitude (mpv) pertains to the wave-length region around 540 nm. Today we can measure magnitudes in the infrared, as well as in the ultraviolet, by using filters in conjunction with the wide spectral sensitivity of photoelectric photometers. So systems of many different magnitudes (colour combinations) are possible. In general, a photometric system requires a detector, filters, and a PH507 Multi-wavelength Professor Michael Smith 15 calibration (in energy units). The properties of the filters are typified by their effective wavelength, 0, and bandpass, ∆ which is defined as the full width at half maximum in the transmission profile. The three main filter types are wide (∆≈ 100 nm), intermediate (∆≈ 10 nm), and narrow (∆≈1 nm). There is a trade-off for the bandwidth choice: a smaller ∆ provides more spectral information but admits less flux into the detector, resulting in longer integration times. For a given range of the spectrum, the design of the filters makes the greatest difference in photometric magnitude systems. A commonly used wide-band magnitude system is the UBV system: a combination of ultraviolet (U), blue (B), and visual (V) magnitudes, developed by H. L. Johnson. These three bands are centred at 365, 440, and 550 nm; each wavelength band is roughly 100 nm wide. In this system, apparent magnitudes are denoted by B or V and the corresponding absolute magnitudes are sub-scripted: MB or MV. To be useful in measuring fluxes, the photometric system must be calibrated in energy units for each of its bandpasses. This calibration turns out to be the hardest part of the job. In general, it relies first on a set of standard stars that define the magnitudes, for a particular filter set and detector; that is, these stars define the standard magnitudes for the photometric system to the precision with which they can be measured. Infrared Windows The UBV system has been extended into the red and infrared (in part because of the development of new detectors, such as CCDs, sensitive to this region of the spectrum). The extensions are not as well standardised as that for the Johnson UBV system, but they tend to include R and I in the far red and J, H, K, L, and PH507 Multi-wavelength Professor Michael Smith 16 M in the infrared. As well as measuring the properties of individual stars at different wavelengths, observing at loner wavelengths, particularly in the infrared, allows us to probe through clouds of small solid dust particles, as seen below A visible-light (left) vs. 2MASS infrared-light (right) view of the central regions of the Milky Way galaxy graphically illustrating the ability of infrared light to penetrate the obscuring dust. The field-of-view is 10x10 degrees Infrared passbands which allow transmission (low absorption): J Band: 1.3 microns H Band: 1.6 microns K band: 2.2 microns L band 3.4 microns M band 5 microns N band 10.2 microns Q band 21 microns Bolometric magnitudes can be converted to total radiant energy flux: One star of Mbol = 0 radiates 2.97 1028 Watts. System is defined by Vega at 7.76 parsecs from the Sun with an apparent magnitude defined as zero. With Lbol = 50.1 Lsolar and Mbol = 0.58. Sun: mbol = -26.8 Full moon: -12.6 Venus: -4.4 Sirius: -1.55 Brightest quasar: 12.8 For Vega: mb = mv = 0. mk = +0.02 Sun: Mb = 5.48, Mv = 4.83, Mk = 3.28 Colour Index: B-V, J-H, H-K are differences in magnitude….flux ratios. PH507 Multi-wavelength Professor Michael Smith 17 But cooler, redder objects possess higher values. Extinction Interstellar Medium modifies the radiation. Dust particles with size of order of the wavelength of the radiation. Blue radiation is strongly scattered compared to red: blue reflection nebulae and reddened stars. Colour Excess: measures the reddening. E(B-V) = B-V - (B-V)o Modified distance modulus: m() = M() + 5 log d – 5 + A() where A () is the extinction due to both scattering and absorption, strongly wavelength dependent. The optical depth is given by exp( ) I Io . Therefore A() = 1.086 The optical depth is where N is the total column density of dust (m-2) between the star and the observer and is the scattering/absorption cross-section (m2). ISM Law related extinction to reddening: Av / E(B-V) = 3.2 + - 0.2 Spectroscopic Parallax Hertzsprung-Russell deduced the main-sequence stars for nearby objects, relating their luminosity to their colour. Groups of distant stars should also\line along the same main-sequence strip. However they appear very dim,\of course due to their distance. On comparison of fluxes, we determine the distance. This works out to about 100,000 pc, beyond which main-sequence stars are too \dim. Cepheids as Standard Candles: The Period-Luminosity Relationship Cepheids show an important connection between period and luminosity: the pulsation period of a Cepheid variable is directly related to its median PH507 Multi-wavelength Professor Michael Smith 18 luminosity. This relationship was first discovered from a study of the variables in the Magellanic Clouds, two small nearby companion galaxies to our Galaxy that are visible in the night sky of the southern hemisphere. To a good approximation, you can consider all stars in each Magellanic Cloud to be at the same distance. Henrietta Leavitt, working at Harvard in 1912, found that the brighter the median apparent magnitude (and so the luminosity, since the stars are the same distance), the longer the period of the Cepheid variable. A linear relationship was found. Harlow Shapley recognised the importance of this period-luminosity (P-L) relation-ship and attempted to find the zero point, for then a knowledge of the period of Cepheid would immediately indicate its luminosity (absolute magnitude). This calibration was difficult to perform because of the relative scarcity of Cepheids and their large distances. None are sufficiently near to allow a trigonometric parallax to be determined, so Shapley had to depend upon the relatively inaccurate method of statistical parallaxes. His zero point was then used to find the distances to many other galaxies. These distances are revised as new and accurate data become available. Right now, some 20 stars whose distances are known reasonably well (because they are in open clusters) serve as the calibrators for the P-L relationship. Further work showed that there are two types of Cepheids, each with its own separate, almost parallel P-L relationship. PH507 Multi-wavelength Professor Michael Smith 19 The classical Cepheids are the more luminous, of Population I, and found in spiral arms. Population II Cepheids, also known as W Virginis stars after their prototype, are found in globular clusters and other Population II systems. Classical Cepheids have periods ranging from one to 50 days (typically five to ten days) and range from F6 to K2 in spectral class. Population II Cepheids vary in period from two to 45 days (typically 12 to 20 days) and range from F2 to G6 in spectral class. Population I and II Cepheids are both regular, or periodic, variables; their change in luminosity with time follows a regular cycle. Cepheids are bright and distinct. They can be used to determine distances to quite distant galaxies, to about 5 Mpc. HST stretched this to 18 Mpc (Virgo cluster). Tully-Fisher Relation In a spiral galaxy, the centripetal force of gas and stars balances the gravitational force: PH507 Multi-wavelength Professor Michael Smith 20 mV2/R = GmM/R2. If they have the same surface brightness ( L/R2 is constant) and the same massto-light ratio (M/L is constant), then L ~ V4. So, provided we can measure V, certain galaxies can be used as standard candles. (determine V through the 21 cm line of atomic hydrogen in the galaxy). Type 1a Supernovae. The peak light output from these supernovae is always about M b = -19.33 +0.25. Therefore we can infer the distance from the inverse square law. Being so bright , they act as standard cadles to large distances: to 1000 Mpc. Why are they standard candles? White dwarfs I binary systems. Material from\ a companion red giant is dumped on the white dwarf surface until the WD reaches a critical mass (Chandrasekhar mass) of 1.4 solar masses. Explosion occurs with fixed rise and fall of luminosity. Other methods: time delay of light rays due to gravitational lensing, cluster size influences Compton scattering of CMB radiation and bremsstrahlung emission (X-rays). Combining, yields the size estimate (Sunyaev-Zeldovich effect). Or, rotational properties of stars with starspots……. New Method? Reverse argument: knowing the Hubble constant is 72 km/s/Mpc, (WMAP result), distances can be found directly from the redshift! PH507 Multi-wavelength Professor Michael Smith Questions How do we scale the solar system? How do we find the distance to gas clouds? PLANET REVIEW The Terrestrial Solar System 21 PH507 Multi-wavelength Professor Michael Smith 22 In the picture above we see the positions of the asteroid belt (green) and other nearearth objects The material in the plane of the Solar System is known as the Kuiper Belt. Surrounding PH507 Multi-wavelength Professor Michael Smith 23 this is a much larger region known as the Oort Cloud, that contains material that occasionally falls in, under the influence of gravity, towards the Sun as comets. The Sun At over 1.4 million kilometers (869,919 miles) wide, the Sun contains 99.86 percent of the mass of the entire solar system: well over a million Earths could fit inside its bulk. The total energy radiated by the Sun averages 383 billion trillion kilowatts, the equivalent of the energy generated by 100 billion tons of TNT exploding each and every second. Planetary configurations • Some of the definitions below make the assumption of coplanar circular orbits. True planetary orbits are ellipses with low eccentricity and inclinations are small so the concepts are applicable in real cases. • Copernicus correctly stated that the farther a planet lies from the Sun, the slower it moves around the Sun. When the Earth and another planet pass each other on the same side of the Sun, the apparent retrograde loop occurs from the relative motions of the other planet and the Earth. PH507 Multi-wavelength Professor Michael Smith 24 As we view the planet from the moving Earth, our line of sight reverses its angular motion twice, and the three-dimensional aspect of the loop comes about because the orbits of the two planets are not coplanar. This passing situation is the same for inferior or superior planets. A Retrograde loop occurs when a superior planet moves through opposition, and occurs as the earth's motion about its orbit causes it to overtake the slower moving superior planet. Thus close to opposition, the planet's motion relative to PH507 Multi-wavelength Professor Michael Smith fixed background stars, follows a small loop. 25 PH507 Multi-wavelength Professor Michael Smith 26 Lecture 4 Mass can be measured in two ways. We could count up the atoms, or count up the molecules and grains of dust and infer the number of atoms. This method can be used if the object is optically thin and we have good tracer: a radiation or scattering mechanism in which the number of photons is related to the number of particles. Otherwise, measuring the mass of an object relies upon its gravitational influence….on nearby bodies or on itself (self-gravity). Newton’s second law states: F = m a, while the first law relates the acceleration to a change is speed or direction. Kepler’s empirical laws for orbital motion thus describe the nature of the acceleration from which masses can be derived. Kepler's Laws First Law: The orbit of each planet is an ellipse with the Sun at one focus p b F r S f q a C Q S = Sun, F = other focus, p = planet. r = HELIOCENTRIC DISTANCE. f = TRUE ANOMALY a = SEMI-MAJOR AXIS = mean heliocentric distance), size of the orbit. b = SEMI-MINOR AXIS. e = ECCENTRICITY, defines shape of orbit. Ellipse: SP + PF = 2a (1) e = CS / a (2) Therefore b2 = a2(1-e2) (3) which defines the PH507 Multi-wavelength •When CS = 0, When CS = , e = 0, e = 1, Professor Michael Smith 27 b = a, the orbit is a circle. the orbit is a parabola. • q = PERIHELION DISTANCE = a - CS = a – ae q = a(1-e) (4) • Q = APHELION DISTANCE = a + CS = a + ae Q = a(1+e) (5) Second Law: For any planet, the radius vector sweeps out equal areas in equal times • Time interval t for planet to travel from p to p1 is the same as time taken for planet to get from p2 to p3. Shaded areas are equal. • Let the time interval t be very small. Then the arc from p to p1 can be regarded as a straight line and the area swept out is the area of the triangle S p p1. If f1 is the angle to p1, and f is the angle to p: p1 p2 p3 r1 p r f S i.e Area = 1/2 r r1 Sin (f1-f). Since t is very small, r ~ r1 and Sin (f1-f) ~ (f1-f) = f Area = 1/2 r2 f The rate this area is swept out is constant according to Kepler's second law, so r2 df/dt = h (6) where h, a constant, is twice the rate of description of area by the radius vector. It is the orbital angular momentum (per unit mass. The total area of the ellipse is πab which is swept out in the orbital period PH507 Multiwavelength 28 P, so using eqn (7) 2ab/P = h. The average angular rate of motion is n = 2/P, so n a2(1-e2)1/2 = h (7) Kepler’s Third Law Kepler's third law took another ten years to develop after the first two. This law relates the period a planet takes to travel around the sun to its average distance and the Sun. This is sometimes called the semi major axis of an elliptical orbit. P 2 = ka3 where P is the period and a is the average distance from the Sun. Kepler’s Third Law follows from the central inverse square nature of the law of gravitation. First look at Newton's law of gravitation - stated mathematically this is F Gm1 m2 r2 Newton actually found that the focus of the elliptical orbits for two bodies of masses m1 and m2 is at the centre of mass. The centripetal forces of a circular orbit are r1 F1 v2 X Centre of M ass m1 m2 v1 F2 r2 2 F1 and m1 v1 4 2 m1 r1 r1 P2 PH507 Multiwavelength 29 2 F2 m2 v2 4 2 m2 r2 r2 P2 where v 2r P and since they are orbiting each other (Newton’s 2nd law) r1 m2 r2 m1 Let's call the separation a = r1 + r2. Then; a r1 m1 r1 m1 r1 1 and multiplying both sides by m2 , am2 m1 r1 m2 r1 m2 m2 or, solving for r1 , r1 am2 m1 m2 Now, since we know that the mutual gravitational force; Fgrav F1 F2 then substituting for r1, 3 a Gm1 m2 2 a G m1 m2 P2 42 Solving for P: P 2 a3 G M1 M2 Third Law is therefore: The cubes of the semi-major axes of the planetary orbits are proportional to the squares of the planets' periods of revolution. Example Europa, one of the Jovian moons, orbits at a distance of 671,000 km from the centre of Jupiter, and has an orbital period of 3.55 days. Assuming that the mass of Jupiter is very much greater than that of Europa, use Kepler's third law to estimate the mass of Jupiter. Using Kepler's third law: PH507 Multiwavelength 30 m jupiter meuropa 4 2 a 3 GP 2 The semi-major axis, a = 6.71 x 105 km = 6.71 x 108 m, and the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds Thus: m jupiter meuropa c 4 2 6.71 108 h 3 19 . 1027 kg c6.67 10 hc3.07 10 h 11 5 2 and since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027 kg. Summary of Kepler’s Laws Summary: Measuring the mass of a planet • Kepler’s third law gives G(M+m) = a3/P2 Since M >> m for all planets, it isn't possible to make precise enough determinations of P and a to determine the masses m of the planets. However, if satellites of planets are observed, then Kepler's law can be used. • Let mp = mass of planet ms = mass of satellite Ps = orbital period of satellite as = semi-major axis of satellite's orbit about the planet. PH507 Then: Multiwavelength 31 G(mp+ms) = 42 as3/Ps2 If the mass of the satellite is small compared with the mass of the planet then mp = 42 as3/(G Ps2) • All the major planets have satellites except Mercury and Venus. Their masses were determined from orbital perturbations on other bodies and later, more accurately from changes in the orbits of spacecraft. So: we can determine the masses of massive objects if we can detect and follow the motion of very low mass satellites. That doesn’t lead very far. How can we determine the masses of distant stars and exoplanets? BASIC STELLAR PROPERTIES - BINARY STARS • For solar type stars, single:double:triple:quadruple system ratios are 45:46:8:1. • Binary nature of stars deduced in a number of ways: VISUAL BINARIES: - Resolvable, generally nearby stars (parallax likely to be available) - Relative orbital motion detectable over a number of years ASTROMETRIC BINARY: only one component detected SPECTROSCOPIC BINARIES: - Unresolved - Periodic oscillations of spectral lines (due to Doppler shift) - In some cases only one spectrum seen SPECTRUM BINARY: 2 sets of lines but no apparent orbital motion but spectrum is clearly combined from stars of differing spectral class. ECLIPSING BINARY: - Unresolved - Stars are orbiting in plane close to line of sight giving eclipses observable as a change in the combined brightness with time (‘’light curves). Some stars may be a combination of these. Visual Binaries • Angular separation ≥ 0.5 arcsec (close to Sun, long orbital periods - years) – example Sirius: PH507 Multiwavelength 32 • Observations: Relative positions: = angular = position Absolute positions: Harder to measure orbits of more massive star A and separation less massive star B about centre of mass C which has proper motion µ. Declination N M otion of centre of mass = proper motion µ Secondary E Right Ascension B Primary C A NB parallax and aberration must also be accounted for. • RELATIVE ORBITS: - TRUE orbit: q = peri-astron distance (arcsec or km) Q = apo-astron distance (arcsec or km) a = semi-major axis (arcsec or km) a = (q + Q)/2 - APPARENT orbits are projected on the celestial sphere Inclination i to plane of sky defines relation between true orbit and apparent orbit. If i≠0° then the centre of mass (e.g. primary) is not at the focus of the elliptical orbit. Measurement of the displacement of the primary gives inclination and true semi-major axis in arcseconds a". PH507 Multiwavelength 33 i i Incline by 45° Apparent orbit True orbit • If the parallax p in arcseconds is observable then a can be derived from a". Earth B radius of Earth's orbit a a" Sun For i=0° p r = distance of binary star a = 1 AU . a"/p" A (In general correction for i≠0 required). Now lets go back to Kepler’s Law … • From Kepler's Law, the Period P is given by 2 3 2 P = 4 a G (mA + mB) (26) For the Earth-Sun system P=1 year, a=1 A.U., mA+mB~msun so 4π2/G = 1 3 a P = (mA + mB) 2 P in years, a in AU, mA,mB in solar masses. From (25) and (26), a" 3 1 mA + mB = ( ) p P2 Sum of masses is determined • ABSOLUTE ORBITS: d c rA A * B rB e f (27) B q A Q PH507 Multiwavelength 34 Semi-major axes aA = (c+e)/2 Minimum separation = q = d + e aB = (d+f)/2 Maximum separation = Q = c + f So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a a = aA + a B (28) (and clearly r =rA + rB) From the definition of centre of mass, mA rA = mB rB ( mA aA = mB aB) mA/mB = aB/aA = rB/rA (29) So from Kepler’s Third Law, which gives the sum of the masses, and Equation (29) above, we get the ratio of masses, ==> mA, mB. Therefore, with both, we can solve for the individual masses of the two stars. Spectroscopic Binaries PH507 Multiwavelength 35 • Orbital period relatively short (hours - months) and i≠0°. • Doppler shift of spectral lines by component of orbital velocity in line of sight (nominal position is radial velocity of system): wavelength wavelength Time Time 2 Stars observable 1 Star observable PH507 Multiwavelength 36 • Data plotted as RADIAL VELOCITY CURVE: recession + v (km s-1) recession + 0 time approach approach 3 2 4 1 3 4 1 3 Observer 4 2 time - ABSOLUTE Relative Orbit 2 v (km s-1) 0 RELATIVE Relative •radial velocity If the orbit iscurve tilted to the line of 2sight (i<90°), the shape is unchanged but velocities are reduced by a factor 1 sin i. v 1 • Take3 a circular orbit with i=90° a = rA + rB v = v A + vB 4 Orbital velocities: 2 vA = 2π rA / P 1 v v = 2π r / P B B 3 1 Since mA rA = mB rB 4 2 mA/mB = rB/rA = vB/vA (31) 1 rB v 1 3 vA r4 1 v = rA v B • Shape of radial velocity curves depends on orbital eccentricity and orientation. • In general, measured velocities are vB sin i and vA sin i, so sin i terms cancel. • From Kepler's law mA + mB = a3/P2 (in solar units). Observed quantities: vA sin i => rA sin i } a sin i vB sin i => rB sin i So can only deduce (mA + mB) sin3 i = (a sin i)3/P2 (32) For a spectroscopic binary, only lower limits to each mass can be derived, unless i is known independently. Eclipsing Binaries • Since stars eclipse i ~ 90° PH507 Astrophysics Professor Glenn White 37 • For a circular orbit: 1, 1' FIRST CONTACT 2, 2' SECOND CONTACT 3, 3' THIRD CONTACT 4, 4' FOURTH CONTACT 4' 3' 2' 1' v 1 2 3 4 Observer in plane • Variation in brightness with time is LIGHTCURVE. • Timing of events gives information on sizes of stars and orbital elements. • Shape of events gives information on properties of stars and relative temperatures. PH507 Multiwavelength 38 Case 1 Smaller star is hotter Case 2 Larger star is hotter F or magnitude Secondary minimum Primary minimum time Case 1 t'1 Case 2 t 1 t'2 t2 t'3 t3 t' 4 t4 t1 t'1 t2 t'2 t3 t'3 t4 t'4 PH507 Multiwavelength 39 • If orbits are circular: minima are symmetrical ie t2-t1 = t4-t3 = t2'-t1' = t4'-t3'; minima are half a period apart; eclipses are of same duration. Assymetrical and/or unevenly spaced minima indicate eccentricity and orientation of orbit. • For a circular orbit: t1 t2 t3 t 4 Distance = velocity x time 2RS = v (t2 - t1) (33) 2RL and 2RS + 2RL = v (t4 - t1) => 2RL = v(t4 - t2) (34) 2RS and ratio of radii RS/RL = (t2 - t1) / (t4 - t2) • Lightcurves are also affected by: Non-total eclises No flat minimum Limb darkening (non-uniform brightness) "rounds off" eclipses Ellipsoidal stars (due to proximity) "rounds off" maxima Reflection effect (if one star is very bright) (35) PH507 Multiwavelength 40 Eclipsing-Spectroscopic Binaries • For eclipsing binaries i ≥ 70° (sin3i > 0.9) • If stars are spectroscopic binaries then radial velocities are known. So from eqns (31) and (32) masses are derived, from eqns (33) and (34) radii are derived, from (36) ratio of temperatures is derived Examine spectra and lightcurve to determine which radius corresponds with which mass and temperature: + v - B From radial velocity curve star A is more massive A Initially A is approaching (blue shift) so first eclipse is A in front of B Since first eclipse is primary eclipse B is hotter than A F If 2 sets of lines are seen then B is larger If 1 set of lines is seen then A is larger time • Since Luminosity L = 4 R2 T4, the ratio of Luminosities is derived from (TO BE DISCUSSED LATER IN COURSE!) LA LB = Summary Type Visual Spectroscopic Eclipsing Eclipsing/ LA/LB spectroscopic RA RB 2 TA 4 TB Observed p, motion on sky Apparent magnitudes Derived a, e, i, mA, mB LA, LB velocity curves lightcurves light + velocity curves MA/MB, (MA+MB)sin3i, a sin i e, i, RS/RL MA, MB, RA, RB, TA/TB, a, e, i, distance LA, LB, TA, T 164PH507 Multiwavelength 41 Lecture 6: Extrasolar Planets 134 other stars are now known to possess planetary systems. 157 planets have been discovered. Although none of the stars has been directly imaged, the effects of the gravity tugging at the stars, as well as the way that gravitation affects can affect material close to the stars, has been clearly seen. Disc of material around the star Beta Pictoris – the image of the bright central star has been artificially blocked out by astronomers using a ‘Coronograph’ • How can we discover extrasolar planets? • Characteristics of the exoplanet population • Planet formation • Explaining the properties of exoplanets Rapidly developing subject - first extrasolar planet around an ordinary star only discovered in 1995 by Mayor & Queloz. Observations thought to be secure, but theory still preliminary... Resources. For observations, a good starting point is Berkeley extrasolar planets search homepage http://exoplanets.org/ Theory: Annual Reviews article by Lissauer (1993) is a good summary of the state of theory prior to the discovery of extrasolar planets Definition of a planet Simplest definition is based solely on mass 164PH507 Multiwavelength 42 • Stars: burn hydrogen (M > 0.075 Msun) • Brown dwarfs: burn deuterium • Planets: do not burn deuterium (M < 0.013 Msun) Deuterium burning limit occurs at around 13 Jupiter masses (1 MJ = 1.9 x 1027 kg ~ 0.001 Msun It is important to realise that for young objects, there is no large change in properties at the deuterium burning limit. ALL young stars / brown dwarfs / planets liberate gravitational potential energy as they contract Types of planet Giant planets (gas giants, `massive’ planets) • Solar System prototypes: Jupiter, Saturn, Uranus... • Substantial gaseous envelopes • Masses of the order of Jupiter mass • In the Solar System, NOT same composition as Sun • Presence of gas implies formation while gas was still prevelant Terrestrial planets • Prototypes: Earth, Venus, Mars • Primarily composed of rocks • In the Solar System (ONLY) orbital radii less than giant planets Much more massive terrestrial planets could exist (>10 Earth masses), though none are present in the Solar System. The Solar system also has asteroids, comets, planetary satellites and rings - we won’t discuss those in this course. Detecting extrasolar planets (1) Direct imaging - difficult due to enormous star / planet flux ratio (2) Radial velocity • Observable: line of sight velocity of star orbiting centre of mass of star - planet binary system • Most successful method so far - all detections to date (3) Astrometry 164PH507 Multiwavelength 43 • Observable: stellar motion in plane of sky • Very promising future method: Keck interferometer, GAIA, SIM (4) Transits • Observable: tiny drop in stellar flux as planet transits stellar disc • Requires favourable orbital inclination • Jupiter mass exoplanet observed from ground HD209458b • Earth mass planets detectable from space (Kepler (2007 launch. NASA Discovery mission), Eddington) (5) Gravitational lensing • Observable: light curve of a background star lensed by the gravitational influence of a foreground star. The light curve shape is sensitive to whether the lensing star is a single star or a binary (star + planet is a special case of the binary) • Rare - requires monitoring millions of background stars, and also unrepeatable • Some sensitivity to Earth mass planets Each method has different sensitivity to planets at various orbital radii - complete census of planets requires use of several different techniques 164PH507 Multiwavelength 44 Planet detection method : Radial velocity technique A planet in a circular orbit around star with semi-major axis a Assume that the star and planet both rotate around the centre of mass with an angular velocity: Using a1 M* = a2 mp and a = a1 + a2, then the stellar speed (v* = a ) in an inertial frame is: (assuming mp << M*). i.e. the stellar orbital speed is small. For a circular orbit, observe a sin-wave variation of the stellar radial velocity, with an amplitude that depends upon the inclination of the orbit to the line of sight: Hence, measurement of the radial velocity amplitude produces a constraint on: mp sin(i) (assuming stellar mass is well-known, as it will be since to measure radial velocity we need exceptionally high S/N spectra of the star). Observable is a measure of mp sin(i). -> given vobs, we can obtain a lower limit to the planetary mass In the absence of other constraints on the inclination, radial velocity searches provide lower limits on planetary masses Magnitude of radial velocity: Sun due to Jupiter: Sun due to Earth: 12.5 m/s 0.1 m/s 164PH507 Multiwavelength 45 i.e. extremely small - 10 m/s is Olympic 100m running pace Spectrograph with a resolving power of 105 will have a pixel scale ~ 10-5 c ~ few km/s Therefore, specialized techniques that can measure radial velocity shifts of ~10-3 of a pixel stably over many years are required High sensitivity to small radial velocity shifts is achieved by: • comparing high S/N = 200 - 500 spectra with template stellar spectra • using a large number of lines in the spectrum to allow shifts of much less than one pixel to be determined. Absolute wavelength calibration and stability over long timescales is achieved by: • passing stellar light through a cell containing iodine, imprinting large number of additional lines of known wavelength into the spectrum • with the calibrating data suffering identical instrumental distortions as the data Error sources: (1) Theoretical: photon noise limit • flux in a pixel that receives N photons uncertain by ~ N1/2 • implies absolute limit to measurement of radial velocity • depends upon spectral type - more lines improve signal • around 1 m/s for a G-type main sequence star with spectrum recorded at S/N=200 • practically, S/N=200 can be achieved for V=8 stars on a 3m class telescope in survey mode (2) Practical: • stellar activity - young or otherwise active stars are not stable at the m/s level and cannot be monitored with this technique • remaining systematic errors in the observations Currently, the best observations achieve: ~ 3 m/s ...in a single measurement. Thought that this error can be reduced to around 1 m/s with further refinements, but not substantially further. The very highest Doppler precisions of 1 m/s are capable\of detecting planets down to about 5 earth masses. Radial velocity monitoring detects massive planets, especially those at small a, but is not sensitive enough to detect Earth-like planets at ~ 1 AU. 164PH507 Multiwavelength 46 Examples of radial velocity data 51 Peg b was the first known exoplanet with a 4 day, circular orbit: a hot Jupiter, \lying close to the central star. Example of a planet with an eccentric orbit: e=0.67 164PH507 Multiwavelength 47 Summary: observables (1) Planet mass, up to an uncertainty from the normally unknown inclination of the orbit. Measure mp sin(i) (2) Orbital period -> radius of the orbit given the stellar mass (3) Eccentricity of the orbit Summary: selection function Need to observe full orbit of the planet: zero sensitivity to planets with P > Psurvey For P < Psurvey, minimum mass planet detectable is one that produces a radial velocity signature of a few times the sensitivity of the experiment (this is a practical detection threshold) Which planets are detectable? m p sin i a 1 2 164PH507 Multiwavelength 48 Current limits: • Maximum a ~ 3.5 AU (ie orbital period ~ 7 years) • Minimum mass ~ 0.5 Jupiter masses at 1 AU, scaling with square root of semimajor axis • No strong selection bias in favour / against detecting planets with different eccentricities Of the first 100 stars found to harbor planets, more than 30 stars host a Jupiter-sized world in an orbit smaller than Mercury's, whizzing around its star in a matter of days. Planet formation is a contest, where a growing planet must fight for survival lest it be swallowed by the star that initially nurtured it. Planet detection method : Astrometry Conceptually identical to radial velocity searches. Light from a planet-star binary is dominated by star. Measure stellar motion in the plane of the sky due to presence of orbiting planet. Must account for parallax and proper motion of star. Magnitude of effect: amplitude of stellar wobble (half peak displacement) for an orbit in the plane of the sky is mp a a1 M* In terms of the angle: m p a M * d for a star at distance d. Note we have again used mp << M* Writing the mass ratio q = mp / M*, this gives: 164PH507 Multiwavelength 49 Note: • Units here are milliarcseconds - very small effect • Different dependence on a than radial velocity method - astrometric planet searches are more sensitive at large a • Explicit dependence on d (radial velocity measurements also less sensitive for distant stars due to lower S/N spectra) • Detection of planets at large orbital radii still requires a search time comparable to the orbital period Detection threshold as function of semi-major axis • Lack of units deliberate! Astrometric detection not yet achieved • As with radial velocity, dependence on orbital inclination, eccentricity • Very promising future: Keck interferometer, Space Interferometry Mission (SIM), ESA mission GAIA, and others • Planned astrometric errors at the ~10 microarcsecond level – good enough to detect planets of a few Earth masses at 1 AU around nearby stars Lecture 7: Planet detection method : Transits Simplest method: look for drop in stellar flux due to a planet transiting across the stellar disc Needs luck or wide-area surveys - transits only occur if the orbit is almost edge-on For a planet with radius rp << R*, probability of transits is: 164PH507 Multiwavelength 50 Close-in planets are more likely to be detected. P = 0.5 % at 1AU, P = 0.1 % at the orbital radius of Jupiter What can we measure from the light curve? (1) Depth of transit = fraction of stellar light blocked This is a measure of planetary radius! In practice, isolated planets with masses between ~ 0.1 MJ and 10 MJ, where MJ is the mass of Jupiter, should have almost the same radii (i.e. a flat mass-radius relation). -> Giant extrasolar planets transiting solar-type stars produce transits with a depth of around 1%. Close-in planets are strongly irradiated, so their radii can be (detectably) larger. But this heating-expansion effect is not generally observed for short-period planets. (2) (3) (4) Duration of transit plus duration of ingress, gives measure of the orbital radius and inclination Bottom of light curve is not actually flat, providing a measure of stellar limbdarkening Deviations from profile expected from a perfectly opaque disc could provide evidence for satellites, rings etc Photometry at better than 1% precision is possible (not easy!) from the ground. HST reached a photometric precision of 0.0001. Potential for efficient searches for close-in giant planets Transit depth for an Earth-like planet is: 164PH507 Multiwavelength 51 Photometric precision of ~ 10-5 seems achievable from space May provide first detection of habitable Earth-like planets NASA’s Kepler mission, ESA version Eddington A reflected light signature must also exist, modulated on the orbital period, even for non-transiting planets. No detections yet. Planet detection method : Gravitational microlensing Light is deflected by gravitational field of stars, compact objects, clusters of galaxies, large-scale structure etc Simplest case to consider: a point mass M (the lens) lies along the line of sight to a more distant source Define: • Observer-lens distance • Observer-source distance • Lens-source distance Azimuthal symmetry -> light from the source appears as a ring ...with radius R0 - the Einstein ring radius - in the lens plane Gravitational lensing conserves surface brightness, so the distortion of the image of the source across a larger area of sky implies magnification. The Einstein ring radius is given by: Dl Ds Dls 164PH507 Multiwavelength 52 Suppose now that the lens is moving with a velocity v. At time t, the apparent distance (in the absence of lensing) in the lens plane between the source and lens is r0. Defining u = r0 / R0, the amplification is: Note: for u > 0, there is no symmetry, so the pattern of images is not a ring and is generally complicated. In microlensing we normally only observe the magnification A, so we ignore this complication... Notes: (1) The peak amplification depends upon the impact parameter, small impact parameter implies a large amplification of the flux from the source star (2) For u = 0, apparently infinite magnification! In reality, finite size of source limits the peak amplification (3) Geometric effect: affects all wavelengths equally (4) Rule of thumb: significant magnification requires an impact parameter smaller than the Einstein ring radius (5) Characteristic timescale is the time required to cross the Einstein ring radius: Optical depth to microlensing Define the optical depth to microlensing as: 164PH507 Multiwavelength 53 This is just the integral of the area of the Einstein ring along the line of sight to the source. For a uniform density of lenses, can easily show that the maximum contribution comes from lenses halfway to the source. Several groups have monitored stars in the Galactic bulge and the Magellanic clouds to detect lensing of these stars by foreground objects (MACHO, Eros, OGLE projects). Original motivation for these projects was to search for dark matter in the form of compact objects in the halo. Timescales for sources in the Galactic bulge, lenses ~ halfway along the line of sight: • Solar mass star ~ 1 month • Jupiter mass planet ~ 1 day • Earth mass planet ~ 1 hour The dependence on M1/2 means that all these timescales are observationally feasible. However, lensing is a very rare event, all of the projects monitor millions of source stars to detect a handful of lensing events. Lensing by a single star Lensing by a star and a planet 164PH507 Multiwavelength 54 What has this to do with planets? Binaries can also act as lenses Light curve for a binary lens is more complicated, but a characteristic is the presence of sharp spikes or caustics. With good enough monitoring, the parameters of the binary doing the lensing can be recovered. Orbiting planet is just a binary with mass ratio q << 1 Planet search strategy: • Monitor known lensing events in realtime with dense, high precision photometry from several sites • Look for deviations from single star lightcurve due to planets • Timescales ~ a day for Jupiter mass planets, ~ hour for Earths • Most sensitive to planets at a ~ R0, the Einstein ring radius • Around 3-5 AU for typical parameters Sensitivity to planets Complementary to other methods: 164PH507 Multiwavelength 55 Actual sensitivity is hard to evaluate: depends upon frequency of photometric monitoring (high frequency needed for lower masses), accuracy of photometry (planets produce weak deviations more often than strong ones) Very roughly: observations with percent level accuracy, several times per night, detect Jupiter, if present, with 10% efficiency Many complicated light curves observed: ...but no strong evidence for planets seen yet RV, Doppler technique (v = 3m/s) 164PH507 Multiwavelength Astrometry: angular oscillation Photometry: transits - close-in planets Microlensing: 56 164PH507 Multiwavelength 57 Direct detection! Photometric : 2005 image of 2M1207 (blue) and its planetary companion, 2M1207b, one of the first exoplanets to be directly imaged, in this case from the Very Large Telescope array in Chile Spectroscopic? The starlight scattered from the planet can be distinguished from the direct starlight because the scattered light is Doppler shifted by virtue of the close-in planet's relatively fast orbital velocity (~ 150 km/sec). Superimposed on the pattern given by the planet's albedo changing slowly with wavelength, the spectrum of the planet's light will retain the same pattern of photospheric absorption lines as in the direct starlight. Lecture 8 The extrasolar planet population Current status of exoplanet searches: Radial Velocity Method (Doppler technique, gravitational wobble) • 156 exoplanets hosted by134 stars discovered, with masses M.sin(i) as low as 6 Earth masses. Generally:~ 0.06 MJ and 10MJ… 164PH507 Multiwavelength 58 164PH507 Multiwavelength 59 and orbital radii from 0.02 AU to 6 AU. • Planet fraction among ~ solar-type stars exceeds 7% • Most are beyond 1 AU • Around 1% of stars have hot Jupiters - massive planets at orbital radii a < 0.1 AU • Four very low mass planets have been detected ….20 earth masses. • Planet occurrence rises rapidly with stellar metallicity • Multiple planets are common, often in resonant orbits Microlensing: two strong detections, low detection rate imply upper limit of ~1/3 on the fraction of lensing stars (~ 0.3 Msun) with Jupiter mass planets at radii to which lensing is most sensitive (1.5 - 4 AU) Transits: 7 known planets (5 found with OGLE photometrically – dimming). Interesting upper limit from non-detection of transits in globular cluster 47 Tuc Transits + Doppler yields mass and size, hence the density of the planet: 0.2 – 1.4 gm/cm3 : mainly gaseous. In addition, sodium and nitrogen found in their atmospheres. Direct Imaging: reports of detections with HST and VLT. Eccentricity: • Except at very small radii, typical planet orbit has significant eccentricity 164PH507 Multiwavelength 60 164PH507 Multiwavelength 61 Eccentricity: Eccentricity vs planet mass 164PH507 Multiwavelength 62 164PH507 Multiwavelength Nothing very striking in these plots: 63 164PH507 Multiwavelength 64 • Accessible region of mp - a space is fully occupied by detected planets • Ignoring the hot Jupiters, no obvious correlation between planet mass and eccentricity... Results from radial velocity searches (1) Massive planets exist at small orbital radii. Closest in planet is at a = 0.035 AU, cf Mercury at ~ 0.4 AU. Less than 10 Solar radii. (2) Hot Jupiters have close to circular orbits. All detected planets with semi-major axis < 0.07 AU have low e. This is similar to binary stars, and is likely due to tidal circularization. (3) Remaining planets have a wide scatter in e, including some planets with large e. Note that the distance of closest approach is a(1-e), and that the effect of tidal torques scales as separation d-6. The very eccentric planet around HD80606 (a = 0.438 AU, e = 0.93, a(1-e) = 0.03 AU) may pose some problems for tidal circularization theory. 164PH507 Multiwavelength 65 Account for this by considering only planets with masses large enough to be detectable at any a < 2.7 AU. -> dN / dlog(a) rises steeply with orbital radius Implies that the currently detected planet fraction ~7% is likely to be a substantial underestimate of the actual fraction of stars with massive planets. Models suggest 15-25% of solar-type stars may have planets with masses 0.2 MJ < mp < 10 MJ. Strong selection effect in favour of detecting planets at small orbital radii, arising from: - lower mass planets can be detected there - mass function rises to smaller masses Observed mass function increases to smaller Mp: 164PH507 Multiwavelength Note: the brown dwarf desert! 66 164PH507 Multiwavelength 67 Constraint from monitoring of 43 microlensing events. Typically, the lenses are low mass stars. At most 1/3 of 0.3 Solar mass stars have Jupiter mass planets between 1.5 AU and 4 AU. Currently consistent with the numbers seen in radial velocity searches - not yet known whether there is a difference in the planet fraction between 0.3 - 1 Solar mass stars. Transit lightcurve from Brown et al. (2001) Consistent with expectations - the probability of a transiting system is ~10%. Measured planetary radius rp = 1.35 J: • Proves we are dealing with a gas giant. • Somewhat larger than models for isolated (non-irradiated) planets effect of environment on structure. • In detail, suggests planet reached its current orbit within a few x 10 Myr after its formation. Precision of photometry with HST / STIS impressive... Metallicity distribution of stars with and without planets 164PH507 Multiwavelength 68 Left plot: metallicity of stars with planets (shaded histogram) compared to a sample of stars with no evidence for planets (open histogram) (data from Santos, Israelian & Mayor, 2001) Host star metallicity Planets are preferentially found around stars with enhanced metal abundance. Cause or effect? High metal abundance could: (a) Reflect a higher abundance in the material which formed the star + protoplanetary disc, making planet formation more likely. (b) Result from the star swallowing planets or planetesimals subsequent to planets forming. If the convection zone is fairly shallow, this can apparently enrich the star with metals even if the primordial material had Solar abundance. Detailed pattern of abundances can distinguish these possibilities, but results currently still controversial. Lack of transits in 47 Tuc A long HST observation monitored ~34,000 stars in the globular cluster 47 Tuc looking for planetary transits. Locally: 1% of stars have hot Jupiters ~ 10% of those show transits Expect 10 -3 x 34,000 ~ few tens of planets None were detected. Possible explanations: 164PH507 Multiwavelength 69 • Low metallicity in cluster prevented planet formation • Cluster environment destroyed discs before planets formed • Stellar fly-bys ejected planets from bound orbits All of these seem plausible - make different predictions for other clusters. 164PH507 Multiwavelength 70 Lecture 9: Radiation processes Almost all astronomical information from beyond the Solar System comes to us from some form of electromagnetic radiation (EMR). We can now detect and study EMR over a range of wavelength or, equivalently, photon energy, covering a range of at least 1016- from short wavelength, high photon energy gamma rays to long wavelength low energy radio photons. Out of all this vast range of wavelengths, our eyes are sensitive to a tiny slice of wavelengths- roughly from 4500 to 6500 Å. The range of wavelengths our eyes are sensitive to is called the visible wavelength range. We will define a wavelength region reaching somewhat shorter (to about 3200 Å) to somewhat longer (about 10,000 Å) than the visible as the optical part of the spectrum. (Note: Physicists measure optical wavelengths in nanometers (nm). Astronomers tend to use _Angstroms. 1 Å = 10-10 m = 0.1 nm. Thus, a physicist would say the optical region extends from 320 to 1000 nm.) All EMR comes in discrete lumps called photons. A photon has a definite energy and frequency or wavelength. The relation between photon energy (Eph) and photon frequency is given by: Eph = h 164PH507 Multiwavelength 71 or, since c = E ph hc where h is Planck’s constant and is the wavelength, and c is the speed of light. The energy of visible photons is around a few eV (electron volts). (An electron volt is a nonmetric unit of energy that is a good size for measuring energies associated with changes of electron levels in atoms, and also for measuring energy of visible light photons. 1 eV = 1.602 x 10-19 Joules.) In purely astronomical terms, the optical portion of the spectrum is important because most stars and galaxies emit a significant fraction of their energy in this part of the spectrum. (This is not true for objects significantly colder than stars - e.g. planets, interstellar dust and molecular clouds, which emit in the infrared or at longer wavelengths - or significantly hotter- e.g. ionised gas clouds, neutron stars, which emit in the ultraviolet and x-ray regions of the spectrum. Another reason the optical region is important is that many molecules and atoms have electronic transitions in the optical wavelength region. Blackbody Radiation Where then does a thermal continuous spectrum come from? Such a continuous spectrum comes from a blackbody whose spectrum depends only upon the absolute temperature. A blackbody is so named because it absorbs all electromagnetic energy incident upon it - it is completely black. To be in perfect thermal equilibrium, however, such a body must radiate energy at exactly the same rate that it absorbs energy; otherwise, the 164PH507 Multiwavelength 72 body will heat up or cool down (its temperature will change). Ideally, a blackbody is a perfectly insulated enclosure within which radiation has come into thermal equilibrium with the walls of the enclosure. Practically, blackbody radiation may be sampled by observing the enclosure through a tiny pinhole in one of the walls. The gases in the interior of a star are opaque (highly absorbent) to all radiation (otherwise, we would see the stellar interior at some wavelength!); hence, the radiation there is blackbody in character. We sample this radiation as it slowly leaks from the surface of the star - to a rough approximation, the continuum radiation from some stars is blackbody in nature. We will define the regions of the Electromagnetic Spectrum to have wavelengthds as follows: Gamma-rays: < 0.1Å, highest frequency, shortest wavelength, highest energy. X-Rays: 0.1Å -- 100Å Ultraviolet light: 100Å -- 3000Å Visible light: 3000Å -- 10000Å = 1µm (micrometer or micron) Infrared Light: 1µm -- 1mm Radio waves: >1mm, lowest frequency, longest wavelength, lowest energy. Planck’s Radiation Law After Maxwell's theory of electromagnetism appeared in 1864, many attempts were made to understand blackbody radiation theoretically. None succeeded until, in 1900, Max K. E. L. Planck (1858-1947) postulated that electromagnetic energy can propagate only in discrete quanta, or photons, each of energy E = hv. 164PH507 Multiwavelength 73 He then derived the spectral intensity relationship, or Planck blackbody radiation law: 2h 3 1 I( )d 2 h c kT e 1 where I(v)dv is the intensity (J/m2 . s . sr) of radiation from a blackbody at temperature T in the frequency range between v and v + dv, h is Planck's constant, c is the speed of light, and k is Boltzmann's constant. Note the exponential in the denominator. Because the frequency v and wavelength of electromagnetic radiation are related by v = c, we may also express Planck's formula in terms of the intensity emitted per unit wavelength interval: This is illustrated for several values of T: 164PH507 Multiwavelength 74 Note that both I() and I(v) increase as the blackbody temperature increases - the blackbody becomes brighter. This effect is easily interpreted when we note that I(v)∆v is directly proportional to the number of photons emitted per second near the energy hv. The Planck function is special enough so that its given its own symbol, B() or B(v), for intensity. Wien’s Law A blackbody emits at a peak intensity that shifts to shorter wavelengths as its temperature increases. 164PH507 Multiwavelength 75 Wilhelm Wien (1864-1928) expressed the wavelength at which the maximum intensity of blackbody radiation is emitted - the peak (that wavelength for which dI()/d = 0) of the Planck curve (found from taking the first derivative of Planck's law) - by Wien's displacement law: max = 2.898 x 10-3 / T where max is in metres when T is in Kelvin. Note that because maxT = constant, increasing one proportionally decreases the other. 164PH507 Multiwavelength 76 For example, the continuum spectrum from our Sun is approximately blackbody, peaking at max ≈ 500 nm. Therefore, the surface temperature is near 5800 K. The Law of Stefan and Boltzmann The area under the Planck curve (integrating the Planck function) represents the total energy flux, F (W/m2), emitted by a blackbody when we sum over all wavelengths and solid angles: 164PH507 Multiwavelength 77 where = 5.669 x 10-8 W/m2 . K4. The strong temperature dependence of this formula was first deduced from thermodynamics in 1879 by Josef Stefan (1835-1893) and was derived from statistical mechanics in 1884 by Boltzmann. Therefore we call the expression the StefanBoltzmann law. The brightness of a blackbody increases as the fourth power of its temperature. If we approximate a star by a blackbody, the total energy output per unit time of the star (its power or luminosity in watts) is just L = 4R2T4 since the surface area of a sphere of radius R is 4R2 To summarise: A blackbody radiator has a number of special characteristics. One, a blackbody emits some energy at all wavelengths. Two, a hotter blackbody emits more energy per unit area and time at all wavelengths than does a cooler one. Three, a hotter blackbody emits a greater proportion of its radiation at shorter wavelengths than does a cooler one. Four, the amount of radiation emitted per second by a unit surface area of a blackbody depends on the fourth power of its temperature. Stellar Material Our Sun is the only star for which I( has been accurately observed. Indeed, Ibol is related to the solar constant: the total 164PH507 Multiwavelength 78 solar radiative flux received at the Earth’s orbit outside our atmosphere (1370 W/m2). The solar luminosity L (3.86 x 1026 W) is calculated from the solar constant in the following manner. Using the inverse-square law, we find the radiative flux at the Sun’s surface R. Then Lis just times this flux. The solar energy distribution curve may be approximated by a Planck blackbody curve at the effective temperature Teff, defined as the temperature of a blackbody that would emit the same total energy as an emitting body, such as the Sun or a star. Then the Stefan-Boltzmann law implies L = 4π R2 T4eff J s-1 where is the Stefan-Boltzmann constant. Stellar Atmospheres The spectral energy distribution of starlight is determined in a star’s atmosphere, the region from which radiation can freely escape. To understand stellar spectra, we first discuss a model stellar atmosphere and investigate the characteristics that determine the spectral features. Physical Characteristics The stellar photosphere, a thin, gaseous layer, shields the stellar interior from view. The photosphere is thin relative to the stellar radius, and so we regard it as a uniform shell of gas. The physical properties of this shell may be approximately specified by the average values of its pressure P, temperature T, and chemical composition µ (chemical abundances). 164PH507 Multiwavelength 79 We also assume that the gas obeys the perfect-gas law: P =nkT where k is Boltzmann’s constant. This relationship is also known as Boyle’s Law. An important result that follows from it is that the kinetic energy of a particle, or assemblage of particles, is given by the relationship; KE 3 kT 2 Thus temperature is just a measure of the kinetic energy of a gas, or an assemblage of particles. This equation applies equally well to a star as a whole, as to a single particle, and later we will look at the comparison between a star’s kinetic and gravitational (potential) energies. The kinetic energy is also a measure of the velocity that atoms or molecules are moving about at - the hotter they are, the faster they move. Thus, for a cloud of gas surrounding a hot star of temperature T = 15,000 K, which consists of hydrogen atoms (mass = 1.67 10-27 kg); 3 1 kT KE mv2 2 2 v 3kT 19 km s 1 50,000 mph m The particle number density is related to both the mass density (kg/m3) and the composition (or mean molecular weight) µ by the following definition of µ: 164PH507 Multiwavelength 80 1 mH n where mH = 1.67 x 10-27 kg is the mass of a hydrogen atom. For a star of pure atomic hydrogen, µ = 1. If the hydrogen is completely ionised, µ = 1/2 because electrons and protons (hydrogen nuclei) are equal in number and electrons are far less massive than protons. In general, stellar interior gases are ionised and 1 3 1 2X Y Z 4 2 where X is the mass fraction of hydrogen, Y is that of helium, and Z is that of all heavier elements. The mass fraction is the percentage by mass of one species relative to the total. Thus, for a pure hydrogen star (X=1.0, Y = 0.0, Z = 0.0), ~ 0.5, and for a white dwarf star (X = 0.0, Y = 1.0, Z = 0.0) ~ 1.33. Temperatures The continuous spectrum, or continuum, from a star may be approximated by the Planck blackbody spectral-energy distribution. For a given star, the continuum defines a colour temperature by fitting the appropriate Planck curve. We can also define the temperature from Wien’s displacement law: maxT = 2.898 x 10-3 m . K which states that the peak intensity of the Planck curve occurs at a wavelength max that varies inversely with the Planck temperature T. The value of max then defines a temperature. Also note here that the hotter a star is, the greater will be its luminous flux (in W/m2), in accordance with the Stefan-Boltzmann law: F = T4 where = 5.67 x 10-8 W/m2 . K4. Then the relation 164PH507 Multiwavelength 81 L = 4πR2T4eff defines the effective temperature of the photosphere. A word of caution: the effective temperature of a star is usually not identical to its excitation (Boltzmann eqn) or ionisation temperature (Saha eqn) because spectral-line formation redistributes radiation from the continuum. This effect is called line blanketing and becomes important when the numbers and strengths of spectral lines are large. When spectral features are not numerous, we can detect the continuum between them and obtain a reasonably accurate value for the star’s effective surface temperature. The line blanketing alters the atmosphere’s blackbody character. Spectrophotometry 164PH507 Multiwavelength 82 The goal of the observational astronomer to to make measurements of the EMR from celestial objects with as much detail, or the finest resolution, possible. There are of course different types of detail that we want to observe. These include angular detail, wavelength detail, and time detail. The perfect astronomical observing system would tell us the amount of radiation, as a function of wavelength, from the entire sky in arbitrarily small angular slices. Such a system does not exist! We are always limited in angular and wavelength coverage, and limited in resolution in angle and wavelength. If we want good information about the wavelength distribution of EMR from an object (spectroscopy or spectrophotometry) we have to give up angular detail. If we want good angular resolution over a wide area of sky (imaging) we usually have to give up wavelength resolution or coverage. The ideal goal of spectrophotometry is to obtain the spectral energy distribution (SED) of celestial objects, or how the energy from the object is distributed in wavelength. We want to measure the amount of energy received by an observer outside the Earth's atmosphere, per second, per unit area, per unit wavelength or frequency interval. Units of spectral flux (in cgs) look like: f = ergs s-1 cm-2 Å -1 if we measure per unit wavelength interval, or f = ergs s-1 cm-2 Hz -1 (pronounced f nu if we measure per unit frequency interval. 164PH507 Multiwavelength 83 Classifying Stellar Spectra Observations A single stellar spectrum is produced when starlight is focused by a telescope onto a spectrometer or spectrograph, where it is dispersed (spread out) in wavelength and recorded photographically or electronically. If the star is bright, we may obtain a high-dispersion spectrum, that is, a few mÅ per millimetre on the spectrogram, because there is enough radiation to be spread broadly and thinly. At high dispersion, a wealth of detail appears in the spectrum, but the method is slow (only one stellar spectrum at a time) and limited to fairly bright stars. Dispersion is the key to unlocking the information in starlight. The Spectral-Line Sequence At first glance, the spectra of different stars seem to bear no relationship to one another. In 1863, however, Angelo Secchi found that he could crudely order the spectra and define different spectral types. Alternative ordering schemes appeared in the ensuing years, but the system developed at the Harvard Observatory by Annie J. Cannon and her colleagues was internationally adopted in 1910. This sequence, the Harvard spectral classification system, is still used today. (About 400,000 stars were classified by Cannon and published in various volumes of the Henry Draper Catalogue, 1910-1924, and its Extension, 1949. At first, the Harvard scheme was based upon the strengths of the hydrogen Balmer absorption lines in stellar spectra, and the spectral ordering was alphabetical (A through to P). Some letters were eventually dropped, and the ordering was rearranged to correspond to a sequence of decreasing temperatures (see the effects of the Boltzmann and Saha equations): OBAFGKMRNS. 164PH507 Multiwavelength 84 Stars nearer the beginning of the spectral sequence (closer to O) are sometimes called early-type stars, and those closer to the M end are referred to as late-type. Each spectral type is divided into ten parts from 0 (early) to 9 (late); for example, . . . F8 F9 G0 G1 G2 . . . G9 K0 . . . . In this scheme, our Sun is spectral type G2. In 1922, the International Astronomical Union (IAU) adopted the Harvard system (with some modifications) as the international standard. Many mnemonics have been devised to help students retain the spectral sequence. A variation of the traditional one is “Oh, Be a Fine Girl, Kiss Me Right Now, Smack.” The next Figure shows exemplary stellar spectra arranged in order; note how the conspicuous spectral features strengthen and diminish in a characteristic way through the spectral types. 164PH507 Multiwavelength 85 Comparison of spectra observed for seven different stars having a range of surface temperatures. The hottest stars, at the top, show lines of helium and multiply-ionised heavy elements. In the coolest stars, at the bottom, helium lines are not seen, but lines of neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen lines are strongest. The actual compositions of all seven stars are about the same. 164PH507 Multiwavelength 86 The Temperature Sequence The spectral sequence is a temperature sequence, but we must carefully qualify this statement. There are many different kinds of temperatures and many ways to deter-mine them. Theoretically, the temperature should correlate with spectral type and so with the star’s colour. From the spectra of intermediate-type stars (A to K), we find that the (continuum) 164PH507 Multiwavelength 87 colour temperature does so, but difficulties occur at both ends of the sequence. For O and B stars, the continuum peaks in the far ultraviolet, where it is undetectable by ground-based observations. Through satellite observations in the far ultraviolet, we are beginning to understand the ultraviolet spectra of O and B stars. For the cool M stars, not only does the Planck curve peak in the infrared, but numerous molecular bands also blanket the spectra of these low-temperature stars. 164PH507 Multiwavelength 88 When the strengths of various spectral features are plotted against excitation-ionisation (or Boltzmann-Saha) temperature; the spectral sequence does correlate with this temperature as seen below; 164PH507 Multiwavelength 89 In practice, we measure a star’s colour index, CI = B - V, to determine the effective stellar temperature. If the stellar continuum is Planckian and contains no spectral lines, this procedure clearly gives a unique temperature, but observational uncertainties and physical effects do lead to problems: (a) for the very hot O and B stars, CI varies slowly with Teff and small uncertainties in its value lead to very large uncertainties in T; (b) for the very cool M stars, CI is large and positive, but these faint stars have not been adequately observed and so CI is not well determined for them; (c) any instrumental deficiencies, calibration errors, or unknown blanketing in the B or V bands affect the value of CI - and thus the deduced T. Hence, it is best to define the CI versus T relation observationally. SPECTROSCOPY • Last year discussed stellar spectra and classification on an empirical basis: Spectral sequence O B A F G K M Temperature ~40,000 K ----> 2500 K Classification based on relative line strengths of He, H, Ca, metal, molecular lines. • We will now look a little deeper at stellar spectra and what they tell us about stellar atmospheres. Radiative Transfer Equation • Imagine a beam of radiation of intensity I passing through a layer of gas: 164PH507 Multiwavelength Power passing into volume 90 Area dA E = I d dA d Power passing out of volume E + dE where I = intensity into solid angle element d path length ds NB in all these equations subscripts can be replaced by In the volume of gas there is: ABSORPTION - Power is reduced by amount dE = - E ds = - I d dA d ds where is the ABSORPTION COEFFICIENT or OPACITY = the cross-section for absorption of radiation of wavelength (frequency ) per unit mass of gas. Units of are m2 kg-1 The quantity is the fraction of power in a beam of radiation of wav absorbed by unit depth of gas. It has units of m-1. (NB in many texts simply given the symbol in the equations given here beware!) EMISSION - Power is increased by amount dE = j d dA d ds (1) where j = EMISSION COEFFICIENT = amount of energy emitted per second per unit mass per unit wavelength into unit solid angle. Units of j (j) are W kg-1 µm-1 sr-1 (W kg-1 Hz-1 sr-1) or m s-3 sr-1 (NB power production per unit volume per unit wavelength into unit so 164PH507 Multiwavelength 91 angle is =j More confusion is possible here, since is also the symbol used for total power output of a gas, units are W kg-1, - Bewar So total change in power is dE = dI d dA d = - I d dA d ds + j d dA d ds which reduces to dI = - I ds + j ds dI ds = - I + j (2) (3) This is a form of the radiative transfer equation in the plane parallel case. Optical depth • Take a volume of gas which only absorbs radiation (j = 0) at : dI = - I ds For a depth of gas s, the fractional change in intensity is given by I (s) dI I I (0) s = 0 - ds ln ( Integrating ==> I (s) I (0) s ) = - ds 0 s - ==> I (s) = I (0) e ds 0 We define Optical depth s ds (4) 164PH507 Multiwavelength 92 - So (5) I (s) = I (0) e • Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its original value if optical depth = 1. • Optical depth is not a physical depth. A large optical depth can occur in a short physical distance if the absorption coefficient is large, or a large physical distance if is small. Full Radiative Transfer Equation again dI = - I + j ds divide by dI ds = -I + j dI d = -I + S As ds --> 0, is constant over ds. This is the RADIATIVE TRANSFER EQUATION in the plane parallel case. Define: S = where j or j = S and S is the SOURCE FUNCTION. Radiative transfer in a blackbody • Remember definition of a blackbody as a perfect absorber and emitter of radiation. Matter and radiation are in THERMODYNAMIC EQUILIBRIUM, ie gross properties do not change with time. Therefore a beam of radiation in a blackbody is constant: (6) 164PH507 Multiwavelength dI ds 93 = 0 = - I + j from definition of source function, j = S ==> 0 = (I - S), i.e. I = S. but for a blackbody I = B the PLANCK FUNCTION 2 B = 2hc 3 1 hc/kT (e - 1) B = 2h 2 c 1 hkT (e - 1) Summary: in complete thermodynamic equilibrium the source function equals the Planck function, i.e. j = B • In studies of stellar atmospheres we make the assumption of LOCAL THERMODYNAMIC EQUILIBRIUM (LTE), i.e. thermodynamic equilibrium for each particular layer of a star. • Note that if incoming radiation at a particular wavelength (e.g. in a spectral line) enters a blackbody gas it is absorbed, but emission is distributed over all wavelengths according to the Planck function. All information about the original energy distribution of the radiation is lost. This is what happens in interior layers of a star where the density is high and photons of any wavelength are absorbed in a very short distance. Such a gas is said to be optically thick (see below). Emission and Absorption lines • the absorption coefficient describes the efficiency of absorption of material in the volume of gas. In a low density gas, photons can generally pass through without interaction with atoms unless they have an energy corresponding to a particular transition (electron energy level transition, or vibrational/rotational state transition in molecules). At this particular (7) (K 164PH507 Multiwavelength 94 energy/frequency/wavelength the absorption coefficient is large. • Let's imagine the volume of gas shown earlier with both absorption and emission: I I (0) path length s dI d = S - I Multiply both sides by e and re-arrange dI ==> d e + I e = S e d ==> d (I e ) = S e integrate over whole volume, i.e. from 0 to s, or 0 to I e ==> = 0 S e 0 assuming S = constant along path I e - I(0) = S e - S ==> ==> I I(0) e- + radiation left over from light entering box. = S (1 - e- ) (8) light from radiation emitted in the box. >> 1: OPTICALLY THICK CASE If >> 1, then e- --> 0, and eqn (8) becomes • Case 1 I = 164PH507 Multiwavelength 95 S In LTE S = B , the Planck function. So for an optically thick gas, the emergent spectrum is the Planck function, independent of composition or input intensity distribution. True for stellar photosphere (the visible "surface" of a star). • Case 2 (9) << 1 OPTICALLY THIN CASE If << 1, then e- ≈ 1 - (first two terms of Taylor series expansion) eqn (8) becomes I = I(0) (1 - ) + S (1 - 1 + ) ==>I = I(0) + ( S - I(0) ) (10) • If I(0) = 0 : no radiation entering the box (from direction of interest): From eqn (8) I = S (= B in LTE) Since = ∫ , then I = s S If is large (true at wavelength of spectral lines) then I is large, we see EMISSION LINES. This happens for example in gaseous nebulae or the solar corona when the sun is eclipsed. • If I(0) ≠ 0 , let's examine eqn (8) I = I(0) + ( S - I(0) ) If S > I(0) then right hand term is +ve when is large (ie is large) we see higher intensity than I(0) ==> EMISSION LINES ON BACKGROUND INTENSITY. If S < I(0) then right hand term is -ve when is large (ie is large) we see lower intensity than I(0) ==>ABSORPTION LINES ON BACKGROUND INTENSITY. For stars we see absorption lines. This means I(0) > S, 164PH507 Multiwavelength 96 i.e. (intensity from deeper layers) > (source function for the top layers Assuming LTE (S = B) the source function increases as temperature increases:I(0) = B(Tdeep layer) > S = B(Touter layer) Therefore temperature must be increasing as we go into the star for absorption lines to be observed. • To summarise - We see CONTINUUM RADIATION for an optically thick gas (= PLANCK FUNCTION assuming LTE). - We see EMISSION LINES for an optically thin gas. - We see ABSORPTION LINES + CONTINUUM for an optically thick gas overlaid by optically thin gas with temperature decreasing outwards. - We see EMISSION LINES + CONTINUUM for an optically thick gas overlaid by an optically thin gas with temperature increasing outwards. Atomic Spectra - Absorption & Emission line series and continua • Bohr theory (last year's physics unit) adequately describes electron energy levels in Hydrogen. Quantum mechanics is required for more massive atoms to describe the dynamics of electrons. However, we are interested here only in the energy levels of electron states rather than a detailed model or description of atomic structure. We can therefore use ENERGY LEVEL DIAGRAMS without 164PH507 Multiwavelength 97 worrying too much about the theory behind them. • There are 3 basic photon absorption mechanisms related to electrons. Using Hydrogen as the example, the electron energy levels, n, are described by E(n) = - 2 2 me e4 Z2 / n2 h2 from Bohr Theory Bound-Bound Transitions • BOUND - BOUND transitions give rise to spectral lines. • ABSORPTION LINE if a photon is absorbed, causing increase in energy of electron. Energy of absorbed photon h = E(nu) - E(nl) (1) where E(nu) and E(nl) are energies of upper and lower energy levels respectively. This is RADIATIVE EXCITATION. • Note energy can also be absorbed from collision of a free particle (COLLISIONAL EXCITATION) - no absorption line is seen in this case. • Atom remains in excited state until SPONTANEOUS EMISSION (photon is emitted typically after ~10-8 s) or INDUCED EMISSION (Photon emitted at same energy and coherently with incoming photon - as in lasers). Both produce EMISSION LINES. • Narrow lines are seen since () is high, otherwise gas transitions can only occur if is transparent and () photon has energy is low. (frequency/wavelength) • Energy level diagram corresponding to difference shows electron energy in energy levels. If this is level changes for the case then the absorption of a photon. absorption coefficient Lowest energy level set to PH507 Astrophysics Dr. S.F. Green zero energy. 1eV = 1.6 x 10-19 J. 98 n=• n=4 n=3 13.6 eV 12.73 eV 12.07 eV n=2 10.19 eV n=1 Lyman Series Balmer Paschen Series Series 0 eV • Series of lines seen -LYMAN SERIES transitions to/from n=1 lines seen in ultraviolet -BALMER SERIES "" n=2 "" visual -PASCHEN SERIES"" n=3 "" infrared ... • Energy of absorbed photon is h = ( - E(nl)) + 1/2 me v2 Bound-free transitions • If photon has energy 1/2 m ev 2 13.6 eV greater than that required n=• 12.73 eV to move an electron in an n=4 12.07 eV atom from its current n=3 energy level to level n=∞, n=2 10.19 eV the electron will be released, ionizing the atom. • Ionization potential for n=1 0 eV Hydrogen is 13.6 e • Since one of the states (free electron) can have any energy, the transition can have any energy and the photon any frequency (above a certain value determined by and E(nl)). Thus BOUND-FREE transitions give an ABSORPTION CONTINUUM. • RE-COMBINATION is a FREE-BOUND transition and PH507 Astrophysics Dr. S.F. Green 99 results in an EMISSION CONTINUUM. • The spectrum produced by absorption from a single continuum • energy level will therefore appear as a series of lines of increasing energy (increasing frequency, decreasing wavelength) up to a limit defined by -E(nl), with an absorption continuum shortward of this limit. • For nl=1 the Lyman series (Lyman-, Lyman- etc) is observed together with the Lyman continuum shortward of =91.2 nm. (Since interstellar space is populated by very low density and low temperature hydrogen (ie with n=1) photons with <91.2nm are easily absorbed so it is opaque in the near-UV). For nl=2 the Balmer series (H, H etc) is observed together with the Balmer continuum shortward of =364.7 nm. Free-free transitions • Absorption of a photon by a free electron in the vicinity of an ion. Electron changes from free energy state with velocity v1 to one with velocity v2 i.e. h = 1/2 me v22 - 1/2 me v12 Determination of • The actual spectrum of a star depends on the physical conditions (notably temperature) and composition of the stellar atmosphere. The intensity is produced at a physical level in the star where ~ 2/3. In order to determine the total spectrum, the value of needs to be determined at all wavelengths. The overall is the sum PH507 Astrophysics Professor Glenn White 100 of the contributions from each atomic/molecular species in the atmosphere. Each component of depends on the number of atoms/molecules with a given energy state capable of absorbing radiation at that frequency and the absorption efficiency. We will deal with the energy state populations first: Boltzmann's equation (Excitation equilibrium) • Boltzmann's equation describes the population distribution of energy states for a particular atom in a gas. The ratio of number of atoms per m3 in energy state B to energy state A: NB NA gB (EA - EB)/kT e gA = (50) where gA and gB are STATISTICAL WEIGHTS (number of different quantum states of the same energy), k = Boltzmann const and T = temperature of gas. NB EB > EA so exponential power is -ve. • The probability of finding an atom in an excited state decreases exponentially with the energy of the excited state, but increases with increasing temperature. Saha Equation (Ionization Equilibrium) • The Boltzmann eqn does not describe all the possible atomic states. Excitation may cause electrons to be lost completely. There are therefore a number of different ionization states for a given atom, each of which has one or more energy states. • The ratio of the number of atoms of ionization state i+1 to those of ionization state i (i=I is neutral, i=II is singly ionized, etc) is given by 3/2 Ni+1 Ni = Ui+1 2 Ui Ne 2 me k T 2 h -i /kT e where Ne is the electron density (number of electrons per PH507 Astrophysics Professor Glenn White 101 m3), i is the ionization potential of the ith ionization state, Ui+1 and Ui are PARTITION FUNCTIONS obtained from • weights: Ui = gi1 + -Ein /kT e in g n=2 the statistical • The higher the Ionization potential, i, the lower the fraction of atoms in the upper ionization state, The higher the Temperature, , the higher the fraction of atoms in the upper ionization state, (Collisional excitation is more likely to ionize atom), The higher the electron density, the lower the fraction of atoms in the upper ionization state (due to recombination). • The Boltzmann and Saha Equations give the fraction of atoms in a given ionization state and energy level allowing (when combined with absorption/emission probabilities) and hence the line strengths to be related to abundances. Example - Abundances in the Sun • In line forming regions in the Sun:T ~ 6000 K, Ne ~ 7x1019 m3. Gas I II UII/UIUIII/UII g1 Hydrogen 13.6 eV 2 2 Calcium 6.1 eV 11.9 eV ~2 ~0.5 1 g2 2 6 From Saha Equation for Hydrogen, the ratio of ionized to un-ionized H NII/NI ≈ 6x10-5 i.e. most of Hydrogen is un-ionized. From Boltzmann equation, ratio of number of atoms with electrons in l n=2 to those in level n=1 (E1-E2 = -10.19 eV) is N2/N1 ≈ 3x10-9 i.e almost all H atoms in ground state. The H Balmer lines which originate from level n=2 are PH507 Astrophysics Professor Glenn White 102 strong only because the H abundance is so high. From Saha Equation for Calcium,NII/NI ≈ 600 and NIII/NII ≈ 2x10 i.e. most of Calcium is in singly ionized state. From Boltzmann equation, ratio of number of atoms with electrons in energy states which contribute to the H and K lines to those in the ground state (E1-E2 = -3.15, -3.13 eV) is (NB/NA)II ≈ 10-2 i.e most Ca atoms in ground state. The H and K lines of Calcium are therefore strong because most Ca atoms in the Sun are in an energy state capable of producing the lines. • For stars cooler than the Sun more H is in the ground state so Balmer lines will be weaker, for stars hotter than the Sun more H is in n=2 state so Balmer lines will be stronger. (T ~ 85000 K needed for N2/N1 =1). But at this temperature NII/NI = 105 so little remains un-ionized. • Balmer line strength depends on excitation (function of T) and ionization (function of T and Ne). Balance of effects occurs at T ~ 10,000 K so Balmer lines are strongest in A0 stars. • A similar effect occurs for other species but at different temperatures. Transition probabilities • Once we know the population of all energy states for a given gaseous species we need to know the transition probabilities for each energy state change before the absorption coefficient can be determined. • The transition probabilities must be calculated from atomic theory or determined by experiment - much time has been invested in this major problem in astrophysics. • The EINSTEIN TRANSITION PROBABILITY (inverse of lifetime): PH507 Astrophysics Professor Glenn White 103 for spontaneous emission, A21 2 for stimulated emission B21 -1 for absorption A12 -1 Total • We can now calculate for a given gaseous species. For Hydrogen (removing spectral line opacities for clarity): Lyman continuum absorption falls off with decreasing due to -1 dependence Log T~25000K (B star) Balmer continuum absorption Paschen continuum absorption T~5000K (G star) (nm) • Similar diagrams exist for other species. The total will be the sum for all species in the star. • The region of a star for which optical depth ~2/3 determines where observed radiation originates. So if is large, then = 2/3 at a high level in the atmosphere and if is low, = 2/3 deep in the atmosphere. Solar photospheric opacity • The solar atmosphere is dominated by hydrogen. The visible surface, the photosphere, has a temperature ~5800 K. However, as can be seen from the diagram above, for hydrogen at low temperatures is low in the visible region (~400-700nm). This is because the continuum absorption in the visible is due to Paschen absorption PH507 Astrophysics Professor Glenn White 104 (electrons originating in level n=3) and most hydrogen is in ground state or n=2 level. We would therefore expect the continuum to come from much deeper in the sun where temperatures are higher. So what causes the high solar photospheric opacity? 500. •The solar opacity comes from the H- ion. The ionization potential for H- --> Log T~25000K (B star) is 0.75 eV (=1650nm). H - bound-free H - free-free From Boltzmann eqn N3/N1 = T~5000K (G star) But from Saha eqn (nm) N(H)/N( Therefore N(H-)/N3 ≈ i.e. number of H- ions is greater than number of H atoms in level n=3, so absorption of photons to dissociate H- to H dominates the continuum absorption in the optical. Limb darkening • The Sun is less bright near the limb than at the centre of the disk. The continuum spectrum of the entire solar disk defines a Stefan-Boltzmann effective temperature of 5800 K for the photosphere, but how does the temperature vary in the photosphere? A clue is evident in a white-light photograph of the Sun. PH507 Astrophysics Professor Glenn White 105 We see that the brightness of the solar disk decreases from the centre to the limb - this effect is termed limb darkening. Limb darkening arises because we see deeper, hotter gas layers when we look directly at the centre of the disk and PH507 Astrophysics Professor Glenn White 106 higher, cooler layers when we look near the limb. Assume that we can see only a fixed distance d through the solar atmosphere. The limb appears darkened as the temperature decreases from the lower to the upper photosphere because, according to the Stefan-Boltzmann PH507 Astrophysics Professor Glenn White 107 law (Section 8-6), a cool gas radiates less energy per unit area than does a hot gas. The top of the photosphere, or bottom of the chromosphere, is defined as height = 0 km. Outward through the photosphere, the temperature drops rapidly then again starts to rise at about 500 km into the chromosphere, reaching very high temperatures in the corona. Formation of solar absorption lines. Photons with energies well away from any atomic transition can escape from relatively deep in the photosphere, but those with energies close to a transition are more likely to be reabsorbed before escaping, so the ones we see on Earth tend to come from higher, cooler levels in the solar atmosphere. The inset shows a close-up tracing of two of the thousands of solar absorption lines, those produced by calcium at about 395 nm. PH507 Astrophysics Professor Glenn White 108 At this point, you may have discerned an apparent paradox: how can the solar limb appear darkened when the temperature rises rapidly through the chromosphere? Answering this question requires an understanding of the concepts of opacity and optical depth. Simply put, the chromosphere is almost optically transparent relative to the photosphere. Hence, the Sun appears to end sharply at its photospheric surface - within the outer 300 km of its 700,000 km radius. Our line of sight penetrates the solar atmosphere only to the depth from which radiation can escape unhindered (where the optical depth is small). Interior to this point, solar radiation is constantly absorbed and re-emitted (and so scattered) by atoms and ions. PH507 Astrophysics Professor Glenn White 109 Y Length of each solid bar is approximately the same, i.e. depth for which =2/3 R y Observer X Rx Since R y > R x, radiation from the edge of the disk, Y, originates from a higher (cooler) region than at the centre of the disk, X. Assuming LTE, the continuum radiation is described by the Planck function since Y is at lower temperature, radiation is of lower intensity Spectral line formation • Lines form higher in atmosphere than continuum. For optical lines this corresponds to lower temperature than continuum and therefore lower intensity (absorption lines) (see p21 where S < I). small ~2/3 low in atmosphere 6500 T (K) high ~2/3 high in atmosphere 4500 0 200 400 km Height above photosphere F PH507 Astrophysics Professor Glenn White 110 Spectral line strength Spectral lines are never perfectly monochromatic. Quantum mechanical considerations govern minimum line width, and many other processes cause line broadening . • For abundance calculations we want to know the total line strength. Total line strength is characterised by EQUIVALENT WIDTH. F Normal line F Fully saturated line Shaded areas are equal = equivalent width Stellar composition • Derived from spectral line strengths in stellar atmospheres. In the solar neighbourhood, the composition of stellar atmospheres is: Element H He C,N,O,Ne,Na,Mg,Al,Si,Ca,Fe, others % mass 70 28 ~2. Spectral line structure • NATURAL WIDTH: Due to uncertainty principle, E=h/t, applied to lifetime of excited state. For "normal" lines the atom is excited (by a photon or collision) to an excited atate which has a short lifetime t ~ 10-8 s. The upper energy level therefore has uncertai energy E and the resultant spectral line (absorption or emission) has an uncertain energy (wavelength). Line has a Lorentz profile, ~ 10-5 nm for visible light. • COLLISIONAL/PRESSURE BROADENING: Outer energy levels of atoms affected by presence of neighbouring charged particles (ions and electrons). random effects lead to line broadening since the energy of PH507 Astrophysics Professor Glenn White 111 upper energy level changes relative to the unexcited state energy level. This is the basis of the Luminosity classification for A,B stars. Gaussian profile. ~ 0.02 - 2 nm. • DOPPLER BROADENING: Due to motions in gas producing the line. Doppler shift occurs for each each photon emitted (or absorbed) since the gas producing the line is moving relative to the observer (or gas producing the photon). Thermal Doppler broadening due to motions of individual atoms in the gas. ~0.01 - 0.02 nm for Balmer lines in the Sun. Gaussian profile. Bulk motions of gas in convection cells. Gaussian profile. • ROTATION: If there is no limb darkening, then lines have hemispherical profile due to combination of radiation from surface elements with different radial velocities. Effect depends on rotation rate, size of star and angle of polar tilt. In general, V sin i is derived from the profile. _ V -1 (km s ) 200 Receding +V A Approaching -V B C F A B o C 100 0 O B A F G K • ATMOSPHERIC OUTFLOW: Many different types. Star with expanding gas shell (result of outburst) gives PCYGNI PROFILE. Continuum (+ absorption lines) from star, emission or PH507 Astrophysics Professor Glenn White 112 absorption lines from shell: F Expanding gas shell B D D C Star D A o Observer B C A C B Radiation from star, A, passes through cooler cloud giving absorption line due to shell material which is blue shifted relative to star. Elsewhere, emission lines are seen. Be STARS: Very rapid rotators with material lost from the equator: Radiation from star, A, passes through cooler cloud giving absorption line. Overall line structure is hemispherical rotation line (B,D). Emission lines seen due to shell material (C,E). C F Rotating gas shell E Star B A o Observer C B A D E D PH507 Astrophysics Professor Glenn White 113 Forbidden lines • Only certain transitions are generally seen for two reasons: 1) Outer energy levels are far from the nucleus so in dense gases, levels are distorted or destroyed by interactions. 2) Selection rules for change of quantum numbers restrict possible transitions. • In fact forbidden transitions are not actually forbidden. However, the probability of a forbidden transition is very low, so an allowed transition will generally occur. The lifetimes in an excited state for which there are no allowed downward transitions are ~10-3 - 109 seconds (ie very low transition probability). These are called METASTABLE STATES. • De-excitation from a metastable state can be by: 1) Collisional excitation, or absorption of another photon to higher energy state allowing another downward transition to the equilibrium state, 2) FORBIDDEN TRANSITION producing a FORBIDDEN LINE. Usually denoted with [], e.g. [OII 731.99]. • Forbidden lines are usually much fainter than those from allowed transitions due to low probability. • In interstellar nebulae excited by UV from nearby hot stars, some elements' excited states have no allowed downward transitions to the ground state. In the absence of frequent collisions (due to low density) or high photon flux, a forbidden transition is the only way to the ground state. • These lines were not understood for a long while. A new element Nebulium was invented to account for them. PH507 Astrophysics Professor Glenn White 114 For instance, the UBV system has about 100 standard stars measured to about ± 0.01 magnitude. Then if we can calibrate the flux of just one of these stars, we have calibrated the system. The calibration is usually given for zero magnitude at each filter; all fluxes are then derived from this base level. The star usually chosen as the calibration star is Vega. Colour index in the BV system. Blackbody curves for 20,000 K and 3000 K, along with their intensities at B and V wavelengths. Note that B - V is negative for the hotter star, positive for the cooler one. Lectures 13-14 : Nearby objects PH507 Astrophysics Professor Glenn White 115 PH507 Astrophysics Professor Glenn White 116 PH507 Astrophysics Professor Glenn White 117 PH507 Astrophysics Professor Glenn White 118 PH507 Astrophysics Professor Glenn White 119 STARS The Hertzsprung-Russell Diagram In 1911, Ejnar Hertzsprung plotted the first such two-dimensional diagram (absolute magnitude versus spectral type) for observed stars, followed (independently) in 1913 by Henry Norris Russell; today, this plot is called a Hertzsprung-Russell (H-R) diagram. We infer the properties of stars from their light - in bulk for fluxes and spread out for spectra. This chapter deals with the wealth of information that can be discerned by studying stellar spectra. First we consider stellar atmospheres, for here the stellar spectra originate. Then we tell the story of spectral observations - how they have been made, correlated, and interpreted. Finally, we present that famous and crucial synthesis - the Hertzsprung-Russell diagram - and some of its implications. This discussion will lead to an understanding of the stars themselves. PH507 Astrophysics Professor Glenn White 120 Most stars have properties within the shaded region known as the main sequence. The points plotted here are for stars lying within about 5 pc of the Sun. The diagonal lines correspond to constant stellar radius, so that stellar size can be represented on the same diagram as luminosity and temperature. (Recall that stands for the Sun.) PH507 Astrophysics Professor Glenn White 121 An H-R diagram for the 100 brightest stars in the sky. Such a plot is biased in favour of the most luminous stars--which appear toward the upper left--because we can see them more easily than we can the faintest stars. PH507 Astrophysics Professor Glenn White 122 PH507 Astrophysics Professor Glenn White 123 Stellar (Main Sequence) Properties With Mass Mass Temp Radius Luminosity 40 MSun 35,000 K 18 RSun 320,000 LSun tMS habitable zone 106 yrs 350-600 AU 17 21,000 8 13,000 107 7 13,500 4 630 8x107 2 8,100 2 20 2x109 1 5,800 1 1 1010 1-2 0.2 2,600 0.32 0.0079 5x1011 0.1-0.2 As will soon become clear to you, this simple diagram represents one of the great observational syntheses in astrophysics. Note that any two of luminosity, magnitude, temperature, and radius could be used, but visual, magnitude, and temperature are universally obtained quantities for stars. Stellar luminosity classes in the H-R diagram. Note that a star's location could be specified by its spectral type and luminosity class instead of by its temperature and luminosity. PH507 Astrophysics Professor Glenn White 124 PH507 Astrophysics Professor Glenn White 125 Magnitude versus Spectral Type The first H-R diagrams considered stars in the solar neighbourhood and plotted absolute visual magnitude, M, versus spectral type, Sp, which is equivalent to luminosity versus spectral type or luminosity versus temperature. The Figure below shows this type of plot for the stars with well-determined distances within about 5 pc of the Sun. Note (a) the well-defined main sequence (class V) with ever-increasing numbers of stars toward later spectral types and an absence of spectral classes earlier than A1 (Sirius), (b) the absence of giants and supergiants (class III and I), and (c) the few white dwarfs at the lower left. In contrast, the H-R diagram for the brightest stars includes a significant number of giants and supergiants as well as several early-type main-sequence stars. Here we have made a selection that emphasises very luminous stars at distances far from the Sun. Note that the H-R diagram of the nearest stars is most representative of those throughout the Galaxy: the most common stars are low-luminosity spectral type M. We can also trace the birth of young stars on this plot: PH507 Astrophysics Professor Glenn White 126 Magnitude versus Colour Because stellar colours and spectral types are roughly correlated, we may construct a plot of absolute magnitude versus colour - called a colour-magnitude diagram. The relative ease and convenience with which colour indices (such as B - V) may be determined for vast numbers of stars dictates the popularity of colour-magnitude plots. The resulting diagrams are very similar to the magnitude-spectral type H-R diagrams considered above. Let’s see what information we can glean from them. The Mass-Luminosity Relationship Just as the determination of the period and size of the Earth’s orbit (by Kepler’s third law) leads to the Sun’s mass, so also have we deduced binary stellar masses. Because it is necessary to know the distance to the binary system in order to establish these masses, we need only observe the radiant flux of each star to find its luminosity. When the observed masses and luminosities for stars in binary systems are plotted, we obtain the correlation called the mass-luminosity relationship. PH507 Astrophysics Professor Glenn White 127 In 1924, Arthur S. Eddington calculated that the mass and luminosity of normal stars like the Sun are related by L M L M His first crude theoretical models indicated that ≈ 3. On a log-log plot, this gives a straight line with a slope of . Main sequence stars do seem to conform to this relationship, although the exponent varies from ≈ 3 for luminous and massive stars through -type stars to 2 for dim red stars of low mass. From a sample of 126 well-studied binary systems, we find that the break in slope below this value is 2.26; above it, 3.99. 3 L ~M The value of the exponent varies for different kinds of stars, and in normally between ~ 2.7 and 4. The value of 3 is appropriate to stars more massive than the sun. PH507 Astrophysics Professor Glenn White 128 The more massive stars burn their fuel very rapidly, leading to short lifetimes: M*/Msun 60 30 10 3 1.5 1 0.1 time (years) 3 million 11 million 32 million 370 million 3 billion 10 billion 1000's billions Spectral type O3 O7 B4 A5 F5 G2 (Sun) M7 For L Mn , value of exponent n 3.9 3.0 2.7 Mass range M M < 7 M 7 M < M < 25 M 25 M < M Today, astrophysical theories of stellar structure explain these results in terms of the different internal structures of stars of different mass and the opacities of stellar atmospheres at different temperatures. Note that the M-L law does not apply to highly evolved stars, such as red giants (with extended atmospheres) and white dwarfs (with degenerate matter. While most stellar masses lie in the narrow range from 0.085M to 100M , stellar luminosities cover the vast span 10-4 ≤ L/L ≤ 106! A useful relationship to give a rule of thumb estimate of a stars surface temperature is; 0.5 M T 5870 M* PH507 Astrophysics Professor Glenn White 129 PH507 Astrophysics Professor Glenn White 130 PH507 Astrophysics Professor Glenn White 131 PH507 Astrophysics Professor Glenn White 132 PH507 Astrophysics Temperature inside the Sun Professor Glenn White 133 PH507 Astrophysics Professor Glenn White 134 PH507 Astrophysics Professor Glenn White 135 Photospheric Temperatures The continuum spectrum of the entire solar disk defines a Stefan-Boltzmann effective temperature of 5800 K for the photosphere, but how does the temperature vary in the photosphere? A clue is evident in a white-light photograph of the Sun. We see that the brightness of the solar disk decreases from the centre to the limb - this effect is termed limb darkening. Limb darkening arises because we see deeper, hotter gas layers when we look directly at the centre of the disk and higher, cooler layers when we look near the limb. PH507 Astrophysics Professor Glenn White 136 Assume that we can see only a fixed distance d through the solar atmosphere. The limb appears darkened as the temperature decreases from the lower to the upper photosphere because, according to the Stefan-Boltzmann law, a cool gas radiates less energy per unit area than does a hot gas. The top of the photosphere, or bottom of the chromosphere, is defined as height = 0 km. Outward through the photosphere, the temperature drops rapidly then again starts to rise at about 500 km into the chromosphere, reaching very high temperatures in the corona. Formation of solar absorption lines. Photons with energies well away from any atomic transition can escape from relatively deep in the photosphere, but those with energies close to a transition are more likely to be reabsorbed before escaping, so the ones we see on Earth tend to come from higher, cooler levels in the solar atmosphere. The inset shows a close-up tracing of two of the thousands of solar absorption lines, those produced by calcium at about 395 nm. PH507 Astrophysics Professor Glenn White 137 At this point, you may have discerned an apparent paradox: how can the solar limb appear darkened when the temperature rises rapidly through the chromosphere? Answering this question requires an understanding of the concepts of opacity and optical depth. Simply put, the chromosphere is almost optically transparent relative to the photosphere. Hence, the Sun appears to end sharply at its photospheric surface within the outer 300 km of its 700,000 km radius. Our line of sight penetrates the solar atmosphere only to the depth from which radiation can escape unhindered (where the optical depth is small). Interior to this point, solar radiation is constantly absorbed and re-emitted (and so scattered) by atoms and ions. PH507 Astrophysics Professor Glenn White 138 The Chromosphere The solar chromosphere extends about 10,000 km above the photosphere, and its gas density is far less than that of the photosphere. This thin layer has a reddish hue - as a result of the Balmer (H) emission of hydrogen - visible during a total solar eclipse. At the limb of the Sun, tenuous jets of glowing gas 500 to 1500 km across extend to a distance of 10,000 km upward from the chromosphere. PH507 Astrophysics Professor Glenn White 139 In the spicules, which are best observed in H, gas is rising at about 20 to 25 km/s. Although spicules occupy less than 1% of the Sun’s surface area and have lifetimes of 15 min or less, they probably play a significant role in the mass balance of the chromosphere, corona, and solar wind, and occur in regions of enhanced magnetic fields PH507 Astrophysics Professor Glenn White 140 Solar spicules, short-lived narrow jets of gas that typically last mere minutes, can be seen sprouting up from the solar chromosphere in this H alpha image of the Sun. The spicules are the thin, dark, spikelike regions. They appear dark against the face of the Sun because they are cooler than the solar photosphere The Corona At solar eclipses, the corona appears as a pearly white halo extending far from the Sun’s limb. A brighter inner halo hugs the solar limb, and coronal streamers extend far into space. PH507 Astrophysics Professor Glenn White 141 The change of gas temperature in the lower solar atmosphere is dramatic. The minimum temperature marks the outer edge of the chromosphere. Beyond that, the temperature rises sharply in the transition zone, finally levelling off at over 1,000,000 K in the corona. Why the Sun's atmosphere is shockingly hot Nobody would have guessed it: the atmosphere of the Sun is much, much hotter than its surface. By more than one million degrees Centigrade in fact. Since 1939, when scientists first determined the temperature of the solar atmosphere - known as the corona - they were unable to come up with a convincing theory of why it greatly exceeded the "mere" 6000° of the visible surface. Nearly 60 years later, SOHO solved the mystery. Once again the MDI acronym (short for Michelson Doppler Imager) is the code needed to decipher the secrets of our nearest star. With MDI, scientists gathered data showing that huge numbers of small, closely intertwined magnetic loops continuously emerge from the Sun's visible surface, clash with one another and dissolve within 40 hours. PH507 Astrophysics Professor Glenn White 142 The loops seem to form a tight pattern that scientists call a magnetic carpet. Their interaction generates electrical and magnetic short-circuits and releases enough energy to heat the corona to temperatures hundreds of times higher than those of the solar surface. The solar atmosphere is permeated with magnetic fields, generated by electrified gas, or plasma, churning violently beneath the visible surface. Solar astronomers have long observed loops of plasma, called coronal loops, which appear to trace out the corona's complex magnetic-field structure, much as iron filings reveal the invisible magnetic field surrounding a magnet. Coronal loops come in various sizes, but most are enormous, capable of spanning several Earths. Solar astronomers know the particles comprising plasma are electrically charged and feel magnetic forces. Thus, scientists thought coronal loops were tubes of plasma trapped by and enclosed in the arch-shaped magnetic fields of the corona. The coronal loops have puzzling features, however. The strong pull of solar gravity led astronomers to believe that the plasma should be dense at the bases of the loop and thin at the top, just as the Earth's gravity pulls our atmosphere close to the surface, causing it to thin with increasing altitude. In fact, coronal loops seem to be about the same density throughout their height, even though some of them extend several hundred thousand miles (over a million kilometers) above the solar surface. PH507 Astrophysics Professor Glenn White 143 Coronal loops come in a variety of shapes and sizes, but most are enormous, capable of spanning several Earth's. Photo: NASA and the TRACE team The Visible Corona Coronal continuum radiation has a temperature of 1 to 2 x 106 K. PH507 Astrophysics Professor Glenn White 144 Photograph of the solar corona during the July, 1991 eclipse, at the peak of the sunspot cycle. At these times, the corona is much less regular and much more extended than at sunspot minimum. Astronomers believe that coronal heating is caused by surface activity on the Sun. The changing shape and size of the corona are the direct result of variations in prominence and flare activity over the course of the solar cycle. PH507 Astrophysics Professor Glenn White 145 PH507 Astrophysics Professor Glenn White 146 Line Emission Forbidden Lines Superimposed on the visible coronal continuum are some emission lines that were unidentified until about 1942, when W. Grotrian of Germany and B. Edlén of Sweden interpreted them. (They were long called “coronium” lines, for they did not fit any known atomic transition). The two strongest lines are the green line of Fe XIV (530.3 nm) and the red line of Fe X (637.4 nm); both are forbidden lines. Two significant obstacles hindered the identification of the coronal emission lines: (1) the responsible transitions are forbidden, and (2) the temperatures of the corona are unexpectedly high. In quantum mechanics, certain energy levels of an atom are metastable because downward transitions from such levels are strongly prohibited. While an ordinary permitted transition takes place in about 10-8 s, these metastable levels may persist for seconds or even days before a forbidden transition occurs. In most laboratory and astrophysical situations, gas densities are so high that collisional de-excitation empties metastable levels very rapidly - there is just not enough time for a forbidden transition to take place. In the near vacuum of the corona, however, metastable levels populated by either photospheric radiation or collisions can decay and forbidden emission features are formed. PH507 Astrophysics Professor Glenn White 147 Solar Activity The Solar Cycle Sunspots Sunspots are photospheric phenomena that appear darker than the surrounding photo-sphere (at about 5800 K) because they are cooler (sunspot continuum temperatures are about 3800 K, and sunspot excitation temperatures are about 3900 K). The darkest, central part (with the temperatures just mentioned) is termed the umbra; the umbra is usually surrounded by the lighter penumbra with its radial filamentary structure. The most important characteristic of a sunspot is its magnetic field. Typical field strengths are near 0.1 T, but fields as strong as 0.4 T have been measured. Related to the magnetic field is a horizontal flow of gas in the sunspot penumbra: gas moves out along the lower filaments and inward along the higher filaments (at speeds up to 6 km/s). A given sunspot has an associated magnetic polarity. Lines of magnetic force diverge from a north magnetic pole and converge at a south pole; you are familiar with this characteristic of bar magnets and our Earth. Two sunspots of complementary polarity are generally found together in a bipolar spot group. The magnetic field in a typical sunspot is about 1000 times greater than the field in neighbouring, undisturbed photospheric regions (which is itself several times stronger than the Earth's field). PH507 Astrophysics Professor Glenn White 148 PH507 Astrophysics Professor Glenn White 149 (a) The looplike structure of this prominence clearly reveals the magnetic field lines connecting the two members of a sunspot pair. (b) This image of a particularly large solar prominence was observed by ultraviolet detectors aboard the Skylab space station in 1979 Sunspot Numbers In 1610, shortly after viewing the sun with his new telescope, Galileo Galilei made the first European observations of Sunspots. Daily observations were started at the Zurich Observatory in 1749 and with the addition of other observatories continuous observations were obtained starting in 1849. The sunspot number is calculated by first counting the number of sunspot groups and then the number of individual sunspots. The "sunspot number" is then given by the sum of the number of individual sunspots and ten times the number of groups. Since most sunspot groups have, on average, about ten spots, this formula for counting sunspots gives reliable numbers even when the observing conditions are less than ideal and small spots are hard to see. Monthly averages (updated monthly) of the sunspot numbers (25 kb GIF image), (37 kb postscript file), (62 kb text file) show that the number of sunspots visible on the sun waxes and wanes with an approximate 11-year cycle. PH507 Astrophysics Professor Glenn White 150 PH507 Astrophysics Professor Glenn White 151 Positional Variation Detailed observations of sunspots have been obtained by the Royal Greenwich Observatory since 1874. These observations include information on the sizes and positions of sunspots as well as their numbers. These data show that sunspots do not appear at random over the surface of the sun but are concentrated in two latitude bands on either side of the equator. A butterfly diagram (142 kb GIF image) (610 kb postscript file) (updated monthly) showing the positions of the spots for each rotation of the sun since May 1874 shows that these bands first form at mid-latitudes, widen, and then move toward the equator as each cycle progresses. The cycles overlap at the time of sunspot cycle minimum with old cycle spots near the equator and new cycle spots at high latitudes. An alternate version of this diagram with different colors for even and odd numbered cycles is available as a 610kb postscript file. The distribution of sunspots in solar latitude varies in a characteristic way during the 11-year sunspot-number cycle. Sunspots tend to reside at high latitudes (±35˚) at the start of a cycle. PH507 Astrophysics Professor Glenn White 152 Most spots are near ±15 at maximum, and the few spots at the end of the cycle cluster near ±8. Very few sunspots are ever found at latitudes greater than ±40. The lifetime of a sunspot ranges from a few days (small spots) to months (large spots). In fact, a sunspot dies at the same latitude where it was born (a characteristic that permits us to determine the solar rotation). What takes place is this: as the cycle progresses, new spots appear at ever lower latitudes. The first high-latitude spots of a cycle appear even before the last low-latitude spots of the previous cycle have vanished. So the sunspots follow active latitude belts during the course of a cycle. Less clear, but just as tantalising, large active regions and spot groups seem to fall into preferred longitudes during a cycle - active longitude belts. Concentrations of magnetic fields appear to persist below the photosphere, so that new spot groups arise from about the same locations as previous ones. The active longitude belts sometimes appear about 180 apart, though not all the time; frequently they are not symmetrical across the equator. PH507 Astrophysics Professor Glenn White 153 They reveal a different persistent pattern in the magnetic field structure in the convective zone. Predictions of sunspot numbers are now routinely made – an excellent web site is www.spaceweather.com The Babcock Model PH507 Astrophysics Professor Glenn White 154 Solar Rotation By observing sunspots with his telescope, Galileo determined that the Sun’s surface rotates eastward (synodically) in about one month. Today, the same method is used in the sunspot zone (other methods, such as Doppler shifts, are necessary above latitude ±40), and we know that the Sun rotates differentially. That is, the rotation period is shorter at the solar equator (about 25d) than at higher latitudes (about 27d at 40 and 30d at 70). To sum up, sunspots reveal on a small scale the complexity and variability of solar magnetic phenomena. The parts of a sunspot are all transient magnetic structures - basically a sheaf of magnetic flux tubes filling the umbra and penumbra and fanning out above them. PH507 Astrophysics Professor Glenn White 155 The Sun: A Model Star Our Sun is the nearest star. The fascinating properties and phenomena of the solar surface layers are easily observed and have been studied intensely. Unfortunately, models for understanding solar phenomena have not kept pace with such detailed data. Because the Sun is a fairly typical star and because it is the only star that spans a large angular diameter as seen from the Earth, the discussion here serves as the physical basis to investigate the other stars. Sun Mass (1024 kg) 1,989,100. 6 3 2 GM (x 10 km /s ) 132,712. Volume (1012 km3) 1,412,000. Volumetric mean radius (km) 696,000. Mean density (kg/m3) 1408. Surface gravity (eq.) (m/s2) 274.0 Escape velocity (km/s) 617.7 Ellipticity 0.00005 Moment of inertia (I/MR2) 0.059 Visual magnitude V(1,0) -26.74 Absolute magnitude Luminosity (1024 J/s) Mass conversion rate (106 kg/s) Mean energy production (10-3 J/kg) Surface emission (106 J/m2s) Spectral type Model values at center of Sun: Central pressure: Central temperature: Central density: Earth 5.9736 0.3986 1.083 6371. 5515. 9.78 11.2 0.0034 0.3308 -3.86 +4.83 384.6 4300. 0.1937 63.29 G2 V (Sun/Earth) 333,000. 333,000. 1,304,000. 109.2 0.255 28.0 55.2 0.015 0.178 - 2.477 x 1011 bar 1.571 x 107 K 1.622 x 105 kg/m3 The Structure of the Sun 1 AU from the Earth with a radius of 6.96 x 105 km (109Ro) and a mass of 1.99 x 1030 kg (333,000M), and luminosity, (rate of total radiative energy output) of 3.86 x 1026 W. The average density of the Sun is only 1400 kg/m3 - consistent with a composition of mostly gaseous hydrogen and helium. From its angular size of about 0.5° and its distance of almost 150 million kilometres, we determine that its diameter is 1,392,000 kilometres (109 Earth diameters and almost 10 times the size of the largest planet, Jupiter). All of the planets orbit the Sun because of its enormous gravity. It has about 333,000 times the Earth's mass and is over 1,000 times as massive as Jupiter. PH507 Astrophysics Professor Glenn White 156 The Sun is made of 94% Hydrogen, 6% Helium, - the other elements make up just 0.13% (the three most abundant ‘metals’ Oxygen, Carbon, and Nitrogen make up 0.11%). PH507 Astrophysics Professor Glenn White 157 This spectacular image of the Sun was made by capturing X rays emitted by our star's most active regions. It was taken by a camera on a rocket lofted shortly before the total solar eclipse of July 1991. (Note the shadow of the Moon approaching from the west, at top.) The brightest regions in all these images have temperatures of about 3 million Kelvin. Lectures Week 7: Star Formation Lectures Week 8: Theory of exoplanets PH507 Astrophysics Professor Glenn White 158 Debris Disks Debris disks are remnant accretion disks with little or no gas left (just dust & rocks), outflow has stopped, the star is visible. Theory: Gas disperses, “planetesimals” form (100 km diameter rocks), collide & stick together due to gravity forming protoplanets (Wetherill & Inaba 2000). Protoplanets interact with dust disks: tidal torques cause planets to migrate inward toward their host stars. Estimated migration time ~ 2 x 105 yrs for Earth-size planet at 5 AU (Hayashi et al. 1985). Perturbations caused by gas giants may spawn smaller planets (Armitage 2000): Start with a stable disk around central star. Jupiter-sized planet forms & clears gap in gas disk. Planet accretes along spiral Disk fragments into more arms, arms become unstable. planetary mass objects. Debris Disks – Outer Disk AB Aurigae outer debris disk nearly face on – see structure & condensations (possible protoplanet formation sites? Very far from star) . (Grady et al. 1999) Two obvious differences between the exoplanets and the giant planets in the Solar System: • Existence of planets at small orbital radii, where our previous theory suggested formation was very difficult. • Substantial eccentricity of many of the orbits. No clear answers to either of these surprises, but lots of ideas... PH507 Astrophysics Professor Glenn White 159 Most conservative possibility: • Planet formation in these extrasolar systems was via the core accretion model ie same as dominant theory for the Solar System • Subsequent orbital evolution modified the planet orbits to make them closer to the star and / or more eccentric We will focus on this option. However, more radical options in which exoplanets form from gravitational instability are also possible. Orbital evolution – migration Need a migration mechanism that can move giant planets from formation at ~5 AU to a range of radii from 0.04 AU upwards. Three theories have been proposed: • Gas disc migration: planet forms within a protoplanetary disc and is swept inwards with the gas as the disc evolves and material accretes onto the star. The most popular theory, as by definition gas must have been present when gas giants form. • Planetesimal disc migration: as above, but planet interacts with a disc of rocks rather than gas. Planet ejects the rocks, loses energy, and moves inwards. • Planet scattering: several massive planets form – subsequent chaotic orbital interactions lead to some (most) being ejected with the survivors moving inwards as above. Gas disc migration Planet interacts with gas in the disc via gravitational force Strong interactions at resonances, eg where disc = nplanet, with n an integer. For example the 2:1 resonance, where n = 2, which lies at 2-2/3 rp = 0.63 rp Resonances at r < rp: Disc gas has greater angular velocity than planet. Loses angular momentum to planet -> moves inwards Resonances at r > rp: Disc gas has smaller angular velocity than planet. Gains angular momentum from planet -> moves outwards PH507 Astrophysics Professor Glenn White 160 Interaction tends to clear gas away from location of planet, Result: planet orbits in a gap largely cleared of gas and dust This process occurs for massive planets (~ Jupiter mass) only - Earth mass planets remain embedded in the gas though gravitational torques can be very important source of orbital evolution for them too. How does this lead to migration? PH507 Astrophysics Professor Glenn White 161 Angular momentum transport in the gas (viscosity) tries to close the gap (recall, diffusive evolution of an accretion disc). Gravitational torques from planet try to open gap wider. Gap edge set by a balance: -> Internal viscous torque = planetary torque Planet acts as a angular momentum ‘bridge’: • Inside gap, outward angular momentum flux transported by viscosity within disc • At gap edge, flux transferred to planet via gravitational torques, then outward again to outer disc • Outside gap, viscosity again operative Typically, gap extends to around the 2:1 resonances interior and exterior to the planet’s orbit. As disc evolves, planet moves within gap like a fluid element in the disc - ie usually inwards. Inward migration time ~ tn = R2 / n ~ few x 105 yr from 5 AU. Mechanism can bring planets in to the hot Jupiter regime. Believe that this mechanism is quantitatively consistent with the distribution of exoplanets at different orbital radii – though the error bars are still very large! PH507 Astrophysics Professor Glenn White 162 Points: data Curve: theoretical migration model Eccentricity Substantial eccentricities of many exoplanets orbits do not have completely satisfactory explanation. Possibilities: (1) Scattering among several massive planets Assumption: planet formation often produces a multiple system which is unstable over long timescales: • Chaotic evolution of a, e (especially e) • Orbit crossing • Eventual close encounters -> ejections • Eccentricity for survivors Advantages: • Given enough planets, close together, definitely works • Can produce very eccentric planets cf e=0.92 example discovered • Some (stable) multiple systems are already known Disadvantages: • Requires planets to form very close together. For two planets, with mass ratios to star q1 = m1 / M*, q2, approximate stability boundary for separations: > 2.4 (q1 + q2)1/3 System is unstable on short timescale if: a2 < a1 (1 + ) For Jupiter mass planets, (1 + ) ~ 1.3 Is it plausible that unstable systems formed in a large fraction of extrasolar planetary systems? • Collisions may produce too many low e systems (2) Disc interactions Assumption: gravitational interaction with disc generates eccentricity PH507 Astrophysics Professor Glenn White 163 Advantages: • Same mechanism as invoked for migration • Works for just one planet • Theoretically, interaction is expected to increase eccentricity if dominated by 3:1 resonance Disadvantages: • Gap is only expected to reach the 3:1 resonance for brown dwarf type masses, not massive planets. Smaller gaps definitely tend to circularize the orbit instead. • Seems unlikely to give very large eccentricities (3) Protoplanetary disc itself is eccentric Assumption: why should discs have circular orbits anyway? Eccentric disc -> eccentric planet? Not yet explored in much depth. A possibility, though again seems unlikely to lead to extreme eccentricities. Scattering theory is currently most popular, possibly augmented by interactions with other planets in resonant orbits. PH507 Astrophysics Professor Glenn White 164