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Electromagnetic radiation Ast 111 Lecture 4a ❑Magnitudes, stellar and solar system objects ❑Photons and light ❑The electro-magnetic spectrum ❑Black bodies ❑Atmospheric absorption Props: CD, hand held spectrographs, laser pointers Brightness of starlight • When we speak of brightness really we are talking of energy flux, f. Flux is energy per unit area per unit time received from the star. (erg cm-2 s-1) • Flux is usually measured over a particular wavelength or frequency range. • Astronomers often use a logarithmic scale to measure brightness. Magnitudes. • Log scale used for Earthquakes (Richter scale) and audio (decibels) Astronomical Magnitudes m f f+ +constant m ==−2.5log 2.5 log constant 10 10 Magnitudesusually are usually defined for flux the flux integrated Magnitude defined for the integrated across a particular wavelength band bands. across particular wavelength B blue 4200Å; V visual which is green, 5300Å; R red 7000Å J, H, K : 1.2,1.6, 2.2 µ m near-IR bands. The difference in magnitudes describes a ratio of fluxes. Difference in magnitudes describes a ratio of fluxes m1 − m2 = −2.5log f1 − 2.5log f 2 m1 m2 = 2.5 log10 f1 + 2.5 log10 f2 f1 f1 = −2.5log = 2.5 log f 2 10 f 2 Magnitudes (continued) m= 2.5 log10 f + constant • The brighter the object the lower than magnitude! • A change of a factor of100 in flux corresponds to a difference of 5 mag • 10% error corresponds to about an uncertainty of 0.1 mag • The faintest star visible by eye is about 6 mag (in V band) • An 8m class telescope will let you observe objects to about 26th mag in V (photometrically) Distance Modulus L f= 4⇡r2 m= Flux f depends on the luminosity L of the star and distance to the star r Flux: Energy per unit area per unit time 2.5 log f + constant magnitude 2.5 log L + 5 log r + constant0 ✓ ◆ r The Distance modulus DM = 5 log10 10pc = Absolute magnitude (objects outside solar system) The magnitude the star would have if it were 10 pc away m = M + DM M absolute magnitude m observed magnitude The Constant …. The constant defined such that the absolute magnitude is zero for an A star at 10pc — in all bands. A star has a particular luminosity and colors. Absolute magnitudes for solar system objects • For objects outside the solar system, the absolute magnitude, M, is that which you would detect if the object was 10pc away. • For solar system objects, the absolute magnitude, H, is the magnitude of the object, at 1AU away, at zero phase angle. • Phase angle is the angle measured at the center of an illuminated body between the light source and the observer. Phase angle So the absolute magnitude, H, is computed as if the object was illuminated by the Sun, was 1AU from the Sun and the observer was sitting on the Sun (toasty). Example with Magnitudes You take a spectrum of a star and find that it has lines typical of a A star and that its colors are mB-mV=0 mB the observed magnitude at B band mV the observed magnitude at V band (consistent with an A star and no extinction!) You measure a magnitude of V=5 Q: How far away is the star? Example with Magnitudes mV=5 observed mag MV=0 Absolute magnitude because it’s an A star DM = 5 The difference is the distance modulus DM = 5 log10 (d/10pc) definition of DM log10(d/10pc) = 1 solve this! d/10pc = 10 d = 100pc distance to star Another Example with Magnitudes A star has an observed magnitude 10 Consider a binary comprised of two of these stars What is the observed magnitude of both stars together? Another Example with Magnitudes A particular star has observed magnitude 10 Consider a binary comprised of two of these stars What is the magnitude of both stars together? mA = -2.5 log10 f + constant = 10.0 m2A = -2.5 log10 (2f) + constant = -2.5 log10 f - 2.5 log10 2 + constant = 10.0 - 2.5 log10 2 = 10.0 - 0.75 = 9.25 The Electromagnetic Spectrum • One of the amazing predictions of Maxwell’s equations is that electric and magnetic fields are related and can propagate as electromagnetic waves. • The speed of light, c, relates the frequency 𝝂 of a photon to its wavelength 𝝀. = c/⌫ • Quantum mechanics then led to the discovery that the light is quantized into photons. Energy E = h𝝂 momentum p = h𝝂/c h is Planck’s constant h = 6.6𝗑10-27 erg s Multi-wavelength astronomy Region Wavelength Frequency Convention for wavelength Convention for flux Radio >1mm <3x11Hz Hz, mm Jy Infrared 700nm-1mm 3x1011-4.3x1014Hz µm Mag or Jy 1-300 µm Visible 400-700nm 4.3x10147.5x1014Hz nm or Ǻ Mag UltraViolet 10-400nm 0.1-100eV 7.5x10143x1016Hz Ǻ or eV erg/cm2/s X-ray 0.1-10nm 0.1-20keV 3x10163x1018Hz keV erg/cm2/s Gamma Ray >20keV >3x1018Hz MeV erg/cm2/s Conventions 1Jy (Jansky) = 10-23 erg cm-2 s-1 Hz-1 used in radio/infrared 1Hz = 1 cycle per second 1ev = 1.6 𝗑 10-12 erg used in X-ray/high energy Magnitude system defined such that an A star has zero absolute magnitude (all bands!) Atmospheric Absorption Transmission windows • It’s difficult and expensive to observe from space. • Most previous astronomy has been done from the ground. • Many filter bands and radio receivers are designed to be restricted to atmospheric windows of transmission. • Atmospheric windows exist (where the atmosphere is transparent) in the radio, submillimeter, near+mid-infrared and optical. • The atmosphere is opaque in the far-infrared, UV, X-ray, and low frequency radio. Astronomy at these wavelengths is done from space. • Note that the atmosphere can also be a source of radiation. Spectra • When we look at an astronomical object, we would like to know how much energy or light it gives of at different wavelengths or frequencies. • The flux as a function of wavelength is known as a spectrum. • We measure the flux per unit wavelength or frequency range. • fv is energy per unit are per unit time per unit frequency range. (erg cm-2 s-1 Hz-1) • fλ is energy per unit are per unit time per unit wavelength range. (erg cm-2 s-1 Å-1 or keV-1 or µm-1) Origin of Continuum, Emission-line, and Absorption Spectra Origin of Continuum, Emission, and Absorption Spectra Spectroscopic Gratings Spectroscopic Gratings Question: Why do spectrographs need a slit? normal incidence Spectroscopic Gratings Why do gratings need a slit? Answer: To restrict angle of incoming light Spectroscopic resolution is reduced with wider slits Demo: CD with different laser pointers, different angles Demo: hand held spectrographs: fluorescent vs sunlight Hydrogen Emission and Absorption The spectrum of hydrogen is important in astronomy because most of the Universe is made of hydrogen. Emission or absorption processes in hydrogen give rise to series, which are sequences of lines corresponding to atomic transitions, each ending or beginning with the same atomic state. The Balmer Series involves transitions starting (for absorption) or ending (for emission) with the first excited state of hydrogen. In visible spectrum. The Lyman Series involves transitions that start or end with the ground state of hydrogen, lines in the UV. Quantum Mechanics • We see in spectra that many systems emit light at particular energies. • Consequently the energy of these systems can only be at particular values; we say the energy is quantized. • The theory of quantum mechanics explains this. Bound Free Radiation • • • If an electron is ejected from a Hydrogen atom, its final energy can take a range of values, depending upon its free momentum. The photon emitted when this happens can be at a range of continuous possible energy values or wavelengths. We refer to this as a boundfree transition. • When an electron jumps from one bound energy level to another (boundbound transitions) then only a few wavelengths are possible. • Spectra can be smooth (continuum) or full of emission lines or can have components of both. Radiation observed at different wavelengths • The energy of the light emitted corresponds to energy changes in the object emitting. • Small changes in energy levels correspond to low energy photons. • Large changes in energy levels correspond to high energy photons. Continuous vs discrete spectrum If there are many possible energy states: Continuous spectrum If there are only a few possible energy states: discrete spectrum Examples of emission processes Wavelength Emission lines Continuum radiation Radio HI spin flip transition Synchrotron radiation Infrared Rotational and Vibrational molecular transitions Dust emission Optical UV Atomic transition lines Atomic transition lines from ionized atoms Atomic transition lines from highly ionized atoms Bound-free, free-free Free-free emission Nuclear exited state transitions, electron positron annihilation Compton scattering X-ray Gamma Ray Free-free Black Bodies • A body is in thermal equilibrium when it freely exchanges energy with its surrounding and a steady state is reached where there is no net energy flow. • To maintain a steady state the body must emit radiation at the same rate that it is absorbed. • A black body is an object that absorbs equally well radiation at all wavelengths. • A black body is described by one parameter, the temperature, T. • The hotter the body, the more energy it gives off. • The shape of the spectrum is something that you can derive in a statistical physics class (Compute the average number of photons in a square cavity per degree of freedom in thermal equilibrium). Black Bodies Described by 1 parameter, the temperature T Thermal Simplest possible description of emission Stefan-Boltzmann law Curves only depend on a single parameter, T Total luminosity L emitted per unit surface area A is proportional to T4 T is temperature The Stefan Boltzmann law: L = T4 A The Stefan Boltzmann constant 𝝈 = 6.7 𝗑 10-5 erg cm-2 K-4 s-1 The luminosity of a sphere with radius R L = 4⇡R2 T 4 frequency Wien Displacement Law The peak intensity occurs at a The peak intensity occurs at wavelength that is inversely a wavelength thattois the inversely proportional proportional temperature to the temperature. λmaxT = 2.90 × 10−1 cm K Relation between temperature and spectrum: X-rays ---- 106K plasma Optical---- 104K ionized gas Radio ---- 100K molecular gas frequency The peak intensity occurs at a wavelength that is inversely proportional to the temperature. λmaxT = 2.90 × 10−1 cm K Wien Displacement Law On a log log plot! Planck’s radiation law 2h⌫ 3 1 B⌫ (⌫, T )d⌫ = 2 c e kh⌫ BT 2hc2 1 B ( , T )d = 5 hc kB T e d⌫ 1 ⌫= c d⌫ = d 1 Units: energy/time/area/solid angle/Hz or energy/time/area/solid angle/ wavelength Spectral radiance, specific intensity frequency c 2 d Planck’s radiation law 3 2h⌫ 1 B⌫ (⌫, T )d⌫ = 2 c e kh⌫ BT d⌫ 1 Triumph of statistical physics h Planck’s constant kB Boltzmann’s constant c speed of light • At large frequencies Wien limit B⌫ / ⌫ 3 e h⌫ kB T frequency d⌫ 1 Wien • At small frequencies (low energy, large ⌫2 B⌫ / wavelengths) T Rayleigh-Jeans limit (to derive expand exponential to first order) Rayleigh-Jeans Planck’s law limits 2h⌫ 3 1 B⌫ (⌫, T )d⌫ = 2 c e kh⌫ BT On a log log plot Rayleigh Jeans power law with slope dependent on temperature Wein limit exponential drop Thermal spectrum Black body assumes emission/absorption at all possible wavelengths Continuum spectrum Stars are well approximated by black bodies Line emission/absorption can be characterized by a gas in thermal equilibrium Emission can still be thermal but not give a continuum spectrum There are also non-thermal emission processes Review • Spectrum, electromagnetic radiation, Hydrogen atom, emission lines, continuous spectrum, atmospheric windows, black body, Planck’s constant. • Magnitudes, Distance Modulus. • Definitions of Absolute Magnitude for solar system objects and for Galactic and extra-galactic objects. • Wien’s displacement law, Stefan-Boltzmann law.