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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)
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Forthe
the sake ofItcomparison,
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fornotice
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is very important
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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.