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Transcript
Photons Observational Astronomy 2017 Part 1 Prof. S.C. Trager Wavelengths, frequencies, and energies of photons Recall that λν=c, where λ is the wavelength of a photon, ν is its frequency, and c is the speed of light in a vacuum, c=2.997925×1010 cm s–1 The human eye is sensitive to wavelengths from ~3900 Å (1 Å=0.1 nm=10–8 cm=10–10 m) – blue light – to ~7200 Å – red light “Optical” astronomy runs from ~3100 Å (the atmospheric cutoff) to ~1 µm (=1000 nm=10000 Å) Optical astronomers often refer to λ>8000 Å as “nearinfrared” (NIR) – because it’s beyond the wavelength sensitivity of most people’s eyes – although NIR typically refers to the wavelength range ~1 µm to ~2.5 µm We’ll come back to this in a minute! The energy of a photon is E=hν, where h=6.626×10–27 erg s is Planck’s constant High-energy (extreme UV, X-ray, γ-ray) astronomers often use eV (electron volt) as an energy unit, where 1 eV=1.602176×10–12 erg Some useful relations: E (erg) 14 ⌫ (Hz) = = 2.418 ⇥ 10 ⇥ E (eV) 1 h (erg s ) c hc 1 1 1 (Å) = = = 12398.4 ⇥ E (eV ) ⌫ ⌫ E (eV) Therefore a photon with a wavelength of 10 Å has an energy of ≈1.24 keV If a photon was emitted from a blackbody of temperature T, then the average photon energy is Eav~kT, where k = 1.381×10–16 erg K–1 = 8.617×10–5 eV K–1 is Boltzmann’s constant. It is sometimes useful to know what frequency corresponds to the average photon energy: h⌫ ⇡ ⌫ (Hz) = T = kT 10 2.08 ⇥ 10 T (K) or 1.44 cm K Note that this wavelength isn’t the peak of the blackbody curve. Consider the blackbody function 2hc 1 B (T ) = 3 exp(hc/ kT ) 1 and assume that λ<hc/kT. Then setting dB (T )/d = 0 we find λpT=0.290 cm K for the peak of the blackbody curve. For the Sun, whose surface temperature is T=5777 K, this implies λp≈5000 Å, or roughly a green color. The relation between energy kT in eV and temperature T in K is particularly useful in high-energy astronomy: 5 kT (eV) = 8.617 ⇥ 10 T (K) T (K) = 1.161 ⇥ 104 kT (eV) Therefore X-rays with a wavelength of 10 Å and an energy of 1.24 keV may have been emitted from a blackbody with a temperature of ~1.4×106 K! The electromagnetic spectrum The electromagnetic spectrum The electromagnetic spectrum Approximate EM bands in astronomy λstart Radio ~1 cm Millimeter 1 mm 10 mm ALMA, JVLA Submillimeter 0.2 mm 1 mm ALMA Infrared 1 µm 0.2 mm near-infrared (NIR) 1 µm 2.5 µm ground-based Ground mid-infrared (MIR) 2.5 µm 25 µm Spitzer, JWST far-infrared (FIR) 25 µm 200 µm (0.2 mm) Herschel Optical 3100 Å 1 µm ground-based, HST visible ~4000 Å ~8000 Å eye Space! Ground Ultraviolet (UV) ~500 Å 3100 Å near-ultraviolet (NUV) 2000 Å 3000–3500 Å GALEX, HST far-ultraviolet (FUV) 900 Å 2000 Å GALEX, HST, FUSE extreme-ultraviolet (EUV) 500 Å 1000 Å EUVE X-ray 0.1 keV (100 Å) 200 kev (0.06 Å) XMM, Chandra γ-ray ~200 keV (0.06 Å) λend Telescopes WSRT, LOFAR @ ~2m Space Fermi, INTEGRAL Ground Band Fluxes, filters, magnitudes, and colors For a point source – like an unresolved star – we can define the spectral flux density S(ν) as the energy deposited per unit time per unit area per unit frequency –1 –2 –1 therefore S(ν) has units of erg s cm Hz The actual energy received by a telescope per second in a frequency band Δν (the bandwidth) is P=Sav(ν)AeffΔν, where Aeff is the effective area of the telescope – which includes effects like telescope obscuration, detector efficiency, atmospheric absorption, etc. – and Sav(ν) is the average spectral flux density over the bandwidth An example: bright radio sources have fluxes of 1.0 Jy (Jansky) at ν=1400 MHz near the 21 cm line of H. Then S(ν)=1×10–23 erg s–1 cm–2 Hz–1 (=1.0 Jy) If we observe a 1 Jy source with a single Westerbork telescope – diameter 25 m, efficiency ≈0.5 at this frequency – with a bandwidth of Δν=1.25 MHz, and assuming Sav(ν)= S(ν) over this bandwidth, the telescope will receive P = ⇡ 1 ⇥ 10 23 3 ⇥ 10 9 1 erg s cm 2 Hz 1 2 6 ⇥ 0.5 ⇥ ⇡(12500 cm) ⇥ 1.25 ⇥ 10 Hz erg s 1 = 3 ⇥ 10 16 W This is a tiny amount of power! It would take ~80% of the age of the Universe to collect enough energy to power a 100W lightbulb for 1 second! In reality, S(ν) and Aeff will (likely) not be constant over the bandwidth Δν, so we should really write Z ⌫2 S(⌫)Ae↵ d⌫ P = ⌫1 The total power flowing across an area is called the flux density F, Z ⌫2 S(⌫)d⌫ F = ⌫1 This is the “Poynting flux” in E&M It has units of erg s–1 cm–2 To find the luminosity, we multiply the flux density over the area of a sphere with a radius equal to the distance between the observer (us!) and the emitting object: r so that L=4πr2F over some bandwidth Δν=ν2–ν1. The luminosity is therefore the total power of an object in some frequency range Δν. Note that we often use the term luminosity to mean the bolometric luminosity, the total power integrated over all frequencies. This definition of luminosity assumes 1. the emission is isotropic – that is, the same in all directions 2. an average spectral flux density over the bandwidth If (2) is incorrect, we should write 2 L = 4⇡r F = 4⇡r 2 Z ⌫2 S(⌫)d⌫ ⌫1 Optical and near-infrared astronomers use magnitudes to describe the intensities of astronomical objects. To define magnitudes, it’s useful to know that NUV– optical–NIR detectors (usually) have a response proportional to the number of photons collected in a given time. We can define a photon spectral flux density Sγ(ν), which is the number of photons (γ) per unit frequency per unit time per unit area. It is simply S(⌫) S (⌫) = h⌫ and has the units s–1 cm–2 Hz–1 The number of photons per unit time and unit area detected is then the photon spectral flux density times an efficiency factor that depends on frequency, integrated over all frequencies: Z 1 F = S (⌫)✏(⌫)d⌫ 0 Here ε(ν) is the efficiency which includes all effects like the filter curve, detector efficiency, absorption and scattering of the telescope, instrument, and atmosphere, etc. Consider two stars with fluxes Fγ(1) and Fγ(2) Then the magnitude difference between these stars is ✓ ◆ F (2) m2 m1 = 2.5 log10 F (1) We use logarithms because human perception of intensity tends to be in logarithmic increments We’ll come back to the zeropoint of this scale shortly! Note that this definition defines the apparent magnitude, the magnitude seen by the detector The coefficient of “–2.5” is important. It says that a ratio of 100 in fluxes (received number of photons) corresponds to a magnitude difference of 5 magnitudes If star 2 is 100 times brighter than star 1, it is 5 magnitudes “brighter” but actually 5 magnitudes less. Confusing, eh? This means that a 1st magnitude (m=1) star is brighter than a 2nd magnitude star (m=2). By how much? Invert our equation for magnitudes: F (2) = 10 F (1) 0.4(m2 m1 ) So if m2–m1=1, then Fγ(2)/Fγ(1)=1/2.512... — a factor of ~2.51 in flux. Some useful properties and “factoids” about magnitudes... The magnitude system is roughly based on natural logarithms: m1 m2 = 0.921 ln(f1 /f2 ) If f 1, then m = m2 m1 1.086 f so the magnitude difference between two objects of nearly-equal brightness is equal to the fractional difference in their brightnesses – i.e., a difference of 0.1 magnitudes is ~10% in brightness A factor of 2 difference in brightness is a difference of 0.75 magnitudes Let’s return to our efficiency term ε(ν): we can write this as ✏(⌫) = f⌫ R⌫ T⌫ where f is the transmission of any filter used to isolate the (frequency) region of interest R is the transmission of the telescope, optics, and detector T is the transmission of the atmosphere (if any) Let’s consider the filter term fν: the transmission of the filter can be chosen as desired (assuming the right materials can be found) so that a specific bandpass can be observed There are many filter systems (see next slide)... Two common filter systems L. Girardi et al.: Isochrones in the SDSS system 207 Fig. 3. The filter sets used in the present work. From top to bottom, we show the filter+detector transmission curves S λ for the systems: (1) HST/NICMOS, (2) HST/WFPC2, (3) Washington, (4) ESO/EMMI, (5) ESO/WFI U BVRIZ + ESO/SOFI JHK, and (6) Johnson-CousinsGlass. All references are given in Sect. 4. To allow a good visualisation of the filter curves, they have been re-normalized to their maximum value of S λ . For the sake of comparison, the bottom panel presents the spectra of Vega (A0V), the Sun (G2V), and a M5 giant, in arbitrary scales of Fλ . 4.3.1. WFI V (ESO#843), R (ESO#844), I (ESO#845), and Z (ESO#846), that – here and in Fig. 3 – are referred to as U BVRIZ for short. The Wide Field Imager (WFI) at the MPG/ESO 2.2 m La Silla Bolometric corrections have been computed in the telescope provides imaging of excellent quality over a 34′ × 33′ field of view. It contains a peculiar set of broad-band filters, VEGAmag system assuming all Vega apparent magnitudes to very different from the “standard” Johnson-Cousins ones. This be 0.03, and in the ABmag system, which is adopted by the EIS can be appreciated in Fig. 3; notice in particular the particular group. The photometric calibration of EIS data is discussed in Arnouts et al. (2001). shapesFig. of 1.the WFI B filter+detector and I filters.transmission Moreover,curves EIS makes useinof The SDSS Sλ adopted this work. They refer to the filter and detector throughputs as seen through the WFI Z filter which does notathave a correspondency airmasses of 1.3 (dashed lines) Apache Point Observatory. in Forthe the sake ofItcomparison, the curvesto fornotice a null airmass lines) are also is very important that any(solid photometric observa- So the (apparent) magnitude difference between two objects is ✓ ◆ F ,B (2) mB (2) mB (1) = B(2) B(1) = 2.5 log10 F ,B (1) where F ,B = Z 1 0 S (⌫)✏B (⌫)d⌫ = Z 1 0 S (⌫)fB R⌫ T⌫ d⌫ We define the color of an object as the magnitude difference of the object in two different filters (“bandpasses”) if the filters are X and Y, then the color (X–Y) is (X Y ) ⌘ mX mY = F 2.5 log F ,X ,Y Most (but not all) magnitude systems are based on taking a magnitude with respect to a star with a known (or predefined) magnitude So to get a “magnitude on system X”, one observes stars with known magnitudes and calibrates the “instrumental magnitudes” onto the “standard system” We’ll discuss this calibration process in great detail later in the course! The Vega system defines a set of A0V stars as having apparent magnitude 0 in all bands of a system The Johnson-Cousins-Glass system is a Vega system, where the magnitudes of all bands in the system are set to 0 for an idealized A0V star at ≈8 pc Another common magnitude zeropoint system is the AB system, in which magnitudes are defined as mAB,⌫ = 2.5 log S(⌫) 48.60 at a given frequency ν; see Fukugita et al. (1995) and Girardi et al. (2002) for more info. Apparent magnitudes depend on the flux of photons received from a source; but this depends on the distance to the source! Remember that L=4πr2F, so for a given L, F∝r–2 To have a measurement of intrinsic luminosity, we must remove this distance dependence. We define the absolute magnitude M to do this: we choose a fiducial distance of 10 pc and define the distance modulus F (r) µ=m M = 2.5 log F (10 pc) ◆ ✓ r = 5 log 10 pc = 5 log r (pc) 5 ...ignoring absorption by dust and cosmological effects. We define the absolute bolometric magnitude as the total power emitted over all frequencies expressed in magnitudes. We set the magnitude scale zeropoint to the (absolute) bolometric magnitude of the Sun, Mbol = 4.74 L + 4.74 Thus Mbol = 2.5 log L 33 1 where L = 3.845 ⇥ 10 erg s Solving for the luminosity of an object, then, we have L = 10 0.4Mbol ⇥ 3.0 ⇥ 1035 erg s 1 independent of the temperature (color) of the source.