Photons Outline

Photons Observational Astronomy 2019 Part 1 Prof. S.C. Trager

2 Outline

Wavelengths, frequencies, and energies of photons The EM spectrum Fluxes, ﬁlters, magnitudes & colors 3 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 “near- infrared” (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! 5

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) = 1 =2.418 10 E (eV) h (erg s ) ⇥ ⇥ c hc 1 1 1 (A)˚ = = = 12398.4 E (eV ) ⌫ ⌫ E (eV) ⇥ Therefore a photon with a wavelength of 10 Å has an energy of ≈1.24 keV 7

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⌫ kT ⇡ ⌫ (Hz) = 2.08 1010T (K) or ⇥ T =1.44 cm K

8 The diference between the AVERAGE and PEAK wavelengths of the blackbody curve comes about because of the “heavy” long wavelength “tail” Note that this wavelength isn’t the peak of the blackbody curve. Consider the blackbody function of the curve. 2hc 1 We see the Sun as white, or, at sunrise/sunset, as yellow because of B (T )= 3 exp(hc/kT) 1 scattering (which we’ll come back to in a later lecture). and assume that λ

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. 9

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! 11-1 Note that the visible/optical range covers <1 decade in wavelength! The electromagnetic Note also that the names and boundaries of the bands are historical and spectrum poorly deﬁned...

11-2 The electromagnetic spectrum 11-3 The electromagnetic spectrum

12 Approximate EM bands in astronomy

Band λstart λend Telescopes Ground Radio ~1 cm WSRT, LOFAR @ ~2m 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 Space! mid-infrared (MIR) 2.5 µm 25 µm Spitzer, JWST far-infrared (FIR) 25 µm 200 µm (0.2 mm) Herschel Ground Optical 3100 Å 1 µm ground-based, HST visible ~4000 Å ~8000 Å eye Ultraviolet (UV) ~500 Å 3100 Å near-ultraviolet (NUV) 2000 Å 3000–3500 Å GALEX, HST Space 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) Fermi, INTEGRAL 13 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 therefore S(ν) has units of erg s–1 cm–2 Hz–1 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 23 1 2 1 P =1 10 erg s cm Hz ⇥ 0.5 ⇡(12500 cm)2 1.25 106 Hz ⇥ 9⇥ 1 ⇥16 ⇥ 3 10 erg s =3 10 W ⇡ ⇥ ⇥ 15

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 ⌫2 P = S(⌫)Ae↵ d⌫ Z⌫1

The total power flowing across an area is called the flux density F, ⌫2 F = S(⌫)d⌫ Z⌫1 This is the “Poynting flux” in E&M

It has units of erg s–1 cm–2 17

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:

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. 19

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⇡r2F =4⇡r2 S(⌫)d⌫ Z⌫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. 21

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:

1 F = S (⌫)✏(⌫)d⌫ Z0 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. 23 note: from here on, I mean log-base-10 for “log” and log-base-e for “ln”

Consider two stars with ﬂuxes Fγ(1) and Fγ(2)

Then the magnitude difference between these stars is F (2) m m = 2.5 log 2 1 10 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? 25

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) 0.4(m m ) = 10 2 1 F (1)

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: m m =0.921 ln(f /f ) 1 2 1 2 If f 1 , then m = m m 1.086f 2 1 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 27 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)... 2 0 4 Some ﬁlter systems in commonL. G use.irard i eFromt al.: I sGirardiochrone setin al.sev 2002,eral pho 2004tometric systems

29 Two common ﬁlter systems

L. Girardi et al.: Isoch rones inth e SDSS sy stem 2 0 7

Fig. 3 . The filter sets used inthe present work. From toptobottom, we show the filter+detector transmissioncurves Sλ for the systems: (1 ) HST/NICMOS, (2 ) HST/WFPC2 , (3 ) Washington, (4 ) ESO/EMMI, (5 ) ESO/WFI UBVRIZ + ESO/SOFI JHK, and (6 ) Johnson-Cousins- Glass. All references are giveninSect. 4 . Toallow a good visualisationof the filter curves, theyhave beenre-normalized totheir maximum value of Sλ. For the sake of comparison, the bottom panel presents the spectra of Vega (A0 V), the Sun(G2 V), and a M5 giant, inarbitrary scales of Fλ.

4 .3 .1 . WFI V (ESO#8 4 3 ), R (ESO#8 4 4 ), I (ESO#8 4 5 ), and Z (ESO#8 4 6 ), that – here and inFig. 3 – are referred toas UBVRIZ 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 qualityover a 3 4 ′ 3 3 ′ field of view. It contains a peculiar set of broad-band fi×lters, VEGAmag system assuming all Vega apparent magnitudes to verydifferent from the “standard” Johnson-Cousins ones. This be 0 .0 3 , and inthe ABmag system, which is adopted bythe EIS canbe appreciated inFig. 3 ; notice inparticular the particular group. The photometric calibrationof EIS data is discussed in Arnouts et al. (2 0 0 1 ). shapesFiogf. 1t.hTehWe SFDISBS fialntedr+Idefiteltcetorrs.trMansomreisosivoenrc,uErvIeSs mSλaakdeopsteudseinothfis work . Th ey refer toth e filter anddetector th rou g h pu ts as seenth rou g h the WaFirImZassfielsteorf 1w.3hi(cdahshdeodelsinneso)tathAavpeacah ecPoorirnetsOpbosnerdveantocry. iFnorththee sak e ofItcoismvpaerrisyonim, tph eorctuarnv etstofornoa tniuclel athiramtaassny(soplihdoltionems)eatreicalosboserva- Johnsopnre-sCenoteuds.iAnsll scyu rsvtesma.re re-normalizedtoth eir max imu m v alu e of Sλ. Tthieobnopttoemrfopramneel dprweseitnhtsWth eFsIptehctarta mofaVkeegsa u(Ase0 Vo)f, tshteanSdu nar(dG2sVta)r,s (e.g. anda M 5 g iant, inarbitrary scales of F . Th e λ scale h ere adoptedis th e same as inFig . 3 of Paper I. Giventhe veryunusual set of filtersλ, the importance of com- Landolt 1 9 9 2 ) toconvert WFI instrumental magnitudes tothe puting isochrones specific for WFI is evident. This has benn standard Johnson-Cousins UBVRI system, will not be in the done sofor the broad WFI filters U (ESO#8 4 1 ), B (ESO#8 4 2 ), WFI VEGAmag system we are dealing with here. Instead, in (wh ere m is a mag nitu de, m0 is a zero-point, andf is th e ph o- requ ires k nowledg e of th e effectiv e th rou g h pu ts in each tonflu x as integ ratedov er a filter pass-band) is replacedby an pass-bandSλ, referring toth e complete instru mental con- inv erse h y perbolic sine fu nction fig u ration(pratical h ints onth is step, reg arding SDSS DR1 data, canbe fou ndinth e URL http://www.sdss.org/ 1 µ( f ) = (m0 2 .5 log b′) asinh − ( f /2 b′) (3 ) dr1/algorithms/fluxcal.html − − ); 3 . conv ert th e ph otonflu x toLu ptonet al. (1 9 9 9 ) modified wh ere a= 2 .5 log e, andb is th e constant (inph otonflu x u nits) ′ mag nitu de scale by means of Eq. (3 ), u sing th e b constant th at g iv es µ(0 ) = m 2 .5 log b for a nu ll flu x . Inpractice, b is ′ 0 ′ ′ ty pical of th e observ ational campaig nu nder consideration. relatedtoth e limitin−g mag nitu de of a g iv enph otometric su r- v ey, andh as tobe fu rnish edtog eth er with th e apparent µ in Of cou rse, th e procedu re is not as simple as one wou ldlik e. any of its data releases. Th is mag nitu de definitionreprodu ces Since at g oodsig nal-to-noise ratios ( f > b) th e Lu ptonet al. th e traditional definitionfor objects measu redwith a sig nal-to- ′ scale coincides with th e classical definitionof mag nitu des, th e noise >5 , av oids problems with neg ativ e flu x es for v ery faint qu estionarises wh eth er it is necessary at all toconv ert models objects, andretains a well-beh av ederror distribu tionfor flu x es toLu ptonet al. (1 9 9 9 ) scale. Infact, for most analy ses of stellar approach ing zero. Hence it is primarily of importance for ob- data it will not be worth wh ile, since one is rarely temptedto jects near th e detectionlimit. deriv e astropy sical qu antities from stars measu redwith larg e It is clear th at th is definitionof mag nitu de is not compatible ph otometric errors. with th e formalism we adopt toderiv e bolometric corrections. For diffu se andfaint objects lik e distant g alax ies, h ow- Actu ally, basic qu antities like bolometric corrections, absolu te ev er, th e situ ationmig h t well be th e opposite one: ev enrela- mag nitu des, and distance modu lu s, cannot be defined in any tiv ely noisy data may containpreciou s astroph y sical informa- simple way if we u se th e Lu pton et al. scale, becau se it is a tion. Usefu l h ints abou t th e dominant stellar popu lations may non-log arith mic one. As a corollary, we cansay th at su ch a 30 resu lt, for instance, from a comparisonbetweenth e integ rated scale represents a conv enient way toex press apparent mag ni- mag nitu des andcolou rs of sing le-bu rst stellar popu lations (pro- tu des andcolou rs near th e su rv ey limit (as demonstratedby v idedinth is paper inth e u su al mag nitu de scale) toth ose of Lu ptonet al. 1 9 9 9 ), bu t represents a complicationif we want faint g alax y stru ctu res from SDSS (g iv eninth e Lu ptonet al. torepresent absolu te mag nitu des. scale). If th is is th e case, th e conv ersionproblem h as to be Considering th is, we donot ev entry toex press ou r th eoret- So the (apparent) magnitudefac edifferenced. between two ical models by means of Lu ptonet al. (1 9 9 9 ) modifiedmag ni- objectstu de scale. W eisdo, h owev er, prov ide a prescriptionof h ow to conv ert absolu te mag nitu des MSλ – g iv enby ou r models inth e 2 .3 . Extin ctio n co efficien ts

AB sy stem – intoanapparent µSλ – as g iv eninth e SDSS data releases. Th is canbe done inth e following way : Th e basic formalism of sy nth etic ph otometry as introdu cedin Paper I, allows aneasy assessment of th e effectFofin,Bterstellar(2) mB(2)1 . conv ert from abBsolu(1)te to app =arent Bmag(2)nitu des u sing tBh e (1)ex tinctio =nonth e ou2tpu.t 5data log. As can10be readily seeninEq. (1 ), u su aldefinitions of distance modu lu s andabsorption, i.e., each stellar spectru m Fλ inou r database canbFe redd,Bened(1)by mSλ = MSλ,0 + (m M)0 + ASλ ; apply ing a g iv enex tinctioncu rv e Aλ, andh ence th e bolometric 2 . conv ert from clas−sical apparent mag nitu des to a ph oton corrections compu tedas u su al. Th e differe✓nce betweenth e BCs ◆ whereflu x , i.e. f = (λ/h c)dλ in th e case of ABmag s; th is deriv edfrom reddenedspectra andth e orig inal (u nreddened) ! 1 1 F,B = S (⌫)✏B(⌫)d⌫ = S (⌫)fBR⌫ T⌫ d⌫ Z0 Z0 31 The “color” of two diferent objects makes no sense…

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

F,X (X Y ) mX mY = 2.5 log ⌘ F,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! 33

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 m = 2.5 log S(⌫) 48.60 AB,⌫ 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 35

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 Thus M = 2.5 log +4.74 bol L 33 1 where L =3.845 10 erg s ⇥ Solving for the luminosity of an object, then, we have

0.4M 35 1 L = 10 bol 3.0 10 erg s ⇥ ⇥ independent of the temperature (color) of the source.