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PH507 Multi-wavelength Professor Michael Smith 1 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 2 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 P, so using eqn (7) PH507 Astrophysics 3 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 F1 r1 v2 X Centre of M ass m1 m2 v1 F2 r2 2 F1 and m1 v1 4 2 m1 r1 r1 P2 2 F2 m2 v2 4 2 m2 r2 r2 P2 PH507 Astrophysics 4 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 r1 or, solving for 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: 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 PH507 Astrophysics 5 the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds Thus: m jupiter meuropa c 4 2 6.71 108 c h 3 hc 6.67 10 11 3.07 105 19 . 1027 kg h 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 Then: ms = mass of satellite Ps = orbital period of satellite as = semi-major axis of satellite's orbit about the planet. G(mp+ms) = 42 as3/Ps2 If the mass of the satellite is small compared with the mass of the planet then PH507 Astrophysics 6 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 Astrophysics 7 • 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 Astrophysics 8 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 Astrophysics 9 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 Astrophysics 10 • 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 Astrophysics 11 • 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 12 • 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 Astrophysics 13 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 Astrophysics 14 • 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 Astrophysics Professor Glenn White 15 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 PH507 Astrophysics Professor Glenn White 16 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 • Stars: burn hydrogen (M > 0.075 Msun) • Brown dwarfs: burn deuterium • Planets: do not burn deuterium (M < 0.013 Msun) PH507 Astrophysics Professor Glenn White 17 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 • 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) PH507 Astrophysics Professor Glenn White 18 (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 PH507 Astrophysics Professor Glenn White 19 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 PH507 Astrophysics Professor Glenn White i.e. extremely small - 20 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. PH507 Astrophysics Professor Glenn White 21 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 Summary: observables PH507 Astrophysics Professor Glenn White 22 (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 PH507 Astrophysics Professor Glenn White 23 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: PH507 Astrophysics Professor Glenn White 24 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 END OF NOTES