Observational Astronomy 2018 Part 1 Prof. S.C. Trager Photons

Photons

Observational Astronomy 2018 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 “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!

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:

Therefore a photon with a wavelength of 10 Å has an energy of ≈1.24 keV

⌫ (Hz) = E (erg)

h (erg s ¹) = 2.418 ⇥ 10¹⁴ ⇥ E (eV) (˚A) = c

⌫ = hc

⌫

1

E (eV) = 12398.4 ⇥ E ¹ (eV ¹)

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

Note that this wavelength isn’t the peak of the blackbody curve. Consider the blackbody function

and assume that λ<hc/kT. Then setting 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.

B (T ) = 2hc

3

1

exp(hc/ kT ) 1

dB (T )/d = 0

The relation between energy kT in eV and temperature T in K is particularly useful in high-energy astronomy:

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!

kT (eV) = 8.617 ⇥ 10 ⁵ T (K) T (K) = 1.161 ⇥ 10⁴ kT (eV)

The electromagnetic

spectrum

The electromagnetic

spectrum

The electromagnetic

spectrum

Band λstart λend Telescopes

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

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

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) Fermi, INTEGRAL

Approximate EM bands in astronomy

Ground Space!GroundSpace

Ground

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

P = 1 ⇥ 10 ²³ erg s ¹ cm ² Hz ¹

⇥ 0.5 ⇥ ⇡(12500 cm)² ⇥ 1.25 ⇥ 10⁶ Hz

⇡ 3 ⇥ 10 ⁹ 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

P =

Z ⌫₂

⌫₁

S(⌫)A_e↵d⌫

The total power flowing across an area is called the flux density F,

This is the “Poynting flux” in E&M It has units of erg s^–1 cm^–2

F =

Z ⌫₂

⌫₁

S(⌫)d⌫

r

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.

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 L = 4⇡r²F = 4⇡r²

Z ⌫₂

⌫₁

S(⌫)d⌫

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

and has the units s^–1 cm^–2 Hz^–1 S (⌫) = S(⌫)

h⌫

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:

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.

F =

Z ₁

0

S (⌫)✏(⌫)d⌫

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

Then the magnitude difference between these stars is

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 m₂ m₁ = 2.5 log₁₀

✓ F (2) F (1)

◆

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 1^st magnitude (m=1) star is brighter than a 2^nd magnitude star (m=2).

By how much? Invert our equation for magnitudes:

So if m2–m1=1, then Fγ(2)/Fγ(1)=1/2.512... — a factor of

~2.51 in flux.

F (2)

F (1) = 10 ^{0.4(m2 m1)}

Some useful properties and “factoids” about magnitudes...

The magnitude system is roughly based on natural logarithms:

If , then

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

m₁ m₂ = 0.921 ln(f₁/f₂)

f 1 m = m₂ m₁ 1.086 f

Let’s return to our efficiency term ε(ν): we can write this as

where

is the transmission of any filter used to isolate the (frequency) region of interest

is the transmission of the telescope, optics, and detector

is the transmission of the atmosphere (if any) f

R

T

✏(⌫) = f_⌫R_⌫T_⌫

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 L. Girardi et al.: Isochrones inseveral photometric systems

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 S_λ. 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

The Wide Field Imager (WFI) at the MPG/ESO 2 .2 m La Silla telescope provides imaging of excellent qualityover a 3 4^′× 3 3^′ field of view. It contains a peculiar set of broad-band filters, verydifferent from the “standard” Johnson-Cousins ones. This canbe appreciated inFig. 3 ; notice inparticular the particular shapes of the WFI B and I filters. Moreover, EIS makes use of the WFI Z filter which does not have a correspondencyinthe Johnson-Cousins system.

Giventhe veryunusual set of filters, the importance of com- puting isochrones specific for WFI is evident. This has benn done sofor the broad WFI filters U (ESO#8 4 1 ), B (ESO#8 4 2 ),

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.

Bolometric corrections have been computed in the VEGAmag system assuming all Vega apparent magnitudes to be 0 .0 3 , and inthe ABmag system, which is adopted bythe EIS group. The photometric calibrationof EIS data is discussed in Arnouts et al. (2 0 0 1 ).

It is veryimportant tonotice that anyphotometric observa- tionperformed with WFI that makes use of standard stars (e.g.

Landolt 1 9 9 2 ) toconvert WFI instrumental magnitudes tothe standard Johnson-Cousins UBVRI system, will not be in the WFI VEGAmag system we are dealing with here. Instead, in

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

Fig. 1 . Th e SDSS filter+detector transmissioncu rv es S_λ adoptedinth is work . Th ey refer toth e filter anddetector th rou g h pu ts as seenth rou g h airmasses of 1 .3 (dash edlines) at Apach e Point Observ atory. For th e sak e of comparison, th e cu rv es for a nu ll airmass (solidlines) are also presented.All cu rv es are re-normalizedtoth eir max imu m v alu e of S_λ. Th e bottom panel presents th e spectra of Veg a (A0 V), th e Su n(G2 V), anda M 5 g iant, inarbitrary scales of F_λ. Th e λ scale h ere adoptedis th e same as inFig . 3 of Paper I.

(wh ere m is a mag nitu de, m0 is a zero-point, andf is th e ph o- tonflu x as integ ratedov er a filter pass-band) is replacedby an inv erse h y perbolic sine fu nction

µ( f ) = (m0 − 2 .5 log b^′) − asinh⁻¹( f /2 b^′) (3 ) wh ere a= 2 .5 log e, andb^′ is th e constant (inph otonflu x u nits) th at g iv es µ(0 ) = m0 −2 .5 log b^′ for a nu ll flu x . Inpractice, b^′ is relatedtoth e limiting 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 any of its data releases. Th is mag nitu de definitionreproduces th e traditional definitionfor objects measu redwith a sig nal-to- noise >5 , av oids problems with neg ativ e flu x es for v ery faint objects, andretains a well-beh av ederror distributionfor flu x es approach ing zero. Hence it is primarily of importance for ob- jects near th e detectionlimit.

It is clear th at th is definitionof mag nitu de is not compatible with th e formalism we adopt toderiv e bolometric corrections.

Actu ally , basic qu antities like bolometric corrections, absolu te mag nitu des, and distance modu lu s, cannot be defined in any simple way if we u se th e Lu pton et al. scale, becau se it is a non-log arith mic one. As a corollary , we can say th at su ch a scale represents a conv enient way toex press apparent mag ni- tu des andcolou rs near th e su rv ey limit (as demonstratedby Lu ptonet al. 1 9 9 9 ), but represents a complicationif we want torepresent absolu te mag nitu des.

Considering th is, we donot ev entry toex press ou r th eoret- ical models by means of Lu ptonet al. (1 9 9 9 ) modifiedmag ni- tu de scale. We do, h owev er, prov ide a prescriptionof h ow to conv ert absolu te mag nitu des M_Sλ – g iv enby ou r models inth e AB sy stem – intoanapparent µ_Sλ – as g iv eninth e SDSS data releases. Th is canbe done inth e following way :

1 . conv ert from absolu te to apparent mag nitu des u sing th e u su al definitions of distance modulu s andabsorption, i.e., m_Sλ = M_Sλ,0 + (m− M)⁰ + A_Sλ;

2 . conv ert from classical apparent mag nitu des to a ph oton flu x , i.e. f = !

(λ/h c)dλ in th e case of ABmag s; th is

requ ires k nowledge of th e effectiv e th rou g h pu ts in each pass-bandSλ, referring toth e complete instru mental con- fig u ration(pratical h ints onth is step, reg arding SDSS DR1 data, canbe fou ndinth e URL http://www.sdss.org/

dr1/algorithms/fluxcal.html);

3 . conv ert th e ph otonflu x to Lu ptonet al. (1 9 9 9 ) modified mag nitu de scale by means of Eq. (3 ), u sing th e b^′ constant ty pical of th e observ ational campaig nu nder consideration.

Of cou rse, th e procedure is not as simple as one wou ldlik e.

Since at g oodsig nal-to-noise ratios ( f > b^′) th e Lu ptonet al.

scale coincides with th e classical definitionof mag nitu des, th e qu estionarises wh eth er it is necessary at all toconv ert models toLu ptonet al. (1 9 9 9 ) scale. Infact, for most analy ses of stellar data it will not be worth wh ile, since one is rarely temptedto deriv e astropy sical qu antities from stars measu redwith larg e ph otometric errors.

For diffu se andfaint objects lik e distant g alax ies, h ow- ev er, th e situ ationmig h t well be th e opposite one: ev enrela- tiv ely noisy data may containpreciou s astroph y sical informa- tion. Usefu l h ints abou t th e dominant stellar popu lations may resu lt, for instance, from a comparisonbetweenth e integ rated mag nitu des andcolou rs of sing le-burst stellar popu lations (pro- v idedinth is paper inth e u su al mag nitu de scale) toth ose of faint g alax y stru ctu res from SDSS (g iv eninth e Lu ptonet al.

scale). If th is is th e case, th e conv ersionproblem h as to be faced.

2 .3 . Extin ctio n co efficien ts

Th e basic formalism of sy nth etic ph otometry as introducedin Paper I, allows aneasy assessment of th e effect of interstellar ex tinctiononth e ou tpu t data. As canbe readily seeninEq. (1 ), each stellar spectru m F_λ inou r database canbe reddenedby apply ing a g iv enex tinctioncu rv e Aλ, andh ence th e bolometric corrections compu tedas u su al. Th e difference betweenth e BCs deriv edfrom reddenedspectra andth e orig inal (u nreddened) Some filter systems in common use. From Girardi et al. 2002, 2004

Two common filter systems

So the (apparent) magnitude difference between two objects is

where

m_B(2) m_B(1) = B(2) B(1) = 2.5 log₁₀

✓ F _,B(2) F _,B(1)

◆

F _,B =

Z ₁

0

S (⌫)✏_B(⌫)d⌫ =

Z ₁

0

S (⌫)f_BR_⌫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 ) ⌘ m^X m_Y = 2.5 log F _,X

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!

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

at a given frequency ν; see Fukugita et al. (1995) and Girardi et al. (2002) for more info.

m_AB,⌫ = 2.5 log S(⌫) 48.60

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

...ignoring absorption by dust and cosmological effects.

µ = m M = 2.5 log

 F (r) F (10 pc)

= 5 log

✓ r 10 pc

◆

= 5 log r (pc) 5

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,

Thus where

Solving for the luminosity of an object, then, we have

independent of the temperature (color) of the source.

M_bol = 4.74 M_bol = 2.5 log L

L + 4.74 L = 3.845 ⇥ 10³³ erg s ¹

L = 10 ^0.4Mbol ⇥ 3.0 ⇥ 10³⁵ erg s ¹