Basic Photometry
Photometry is the art of measuring the brightness of an astronomical object.
In principle, it is straightforward; in practice (as in much of astronomy),
there are many subtleties that can cause you great pains. For this lab, you
do not need to deal with most of these effects, but you should know about them
(especially extinction).
Among the details you should be acquainted with are:
Your detector measures the flux S in a bandpass dw. The
units of the detected flux S are erg/cm^{2}/s/A.
Astronomers use a number of standard bandpasses. You can get an idea of the
number of bandpasses, standard and non-standard, by checking the
CTIO
Filter List.
The bandpasses are set by the
detector response, the filter response, the telescope reflectivity, and,
in some cases, by the transmission
of the atmosphere.
The filters are often merely combinations of Schott
colored glass. Because no two telescope/detector combinations are exactly the
same, one must derive transformations from your particular instrumental
system to the standard system (e.g., Taylor, ApJ Supp 60, 577, 1986).
There are 3 basic types of filters:
- long pass. These transmit light longward of some fiducial wavelength.
The name of the filter often gives the 50% transmission wavelength in nm.
For example, the GG495 filter transmits 50% of the incident light at
495 nm (4950 Angstroms), and has higher transmission at longer wavelengths.
- short pass. These transmit light shortward of some fiducial
wavelength. Examples are CuSO_{4} and BG38. Long-pass and
short-pass filters are used for order-sorting.
- isolating. These filters are used to select particular bandpasses.
They may be made of a combination of long- and short-pass filters. Narrow
band filters are generally interference filters.
Common bands include:
Broad bands
These representative glass combinations are those used at the
Palomar
Observatory 60" telescope. The Johnson U,B,V bands are standard. Johnson
R and I are broader and redward of the Kron-Cousins R and I bands.
The Johnson system was defined by Johnson & Morgan (ApJ 117, 313, 1953)
and by Johnson (Ann Rev Astron Astrophys 4, 193 (1966). Bessel
(PASP 91, 589 (1979) defined the Counsins system R_{C},
I_{C}.
A recent
review of the UBVRI bandpasses is given by M.S. Bessell (PASP, 102, 1181,
1990). We have a set of standard UBVRI filters for the ST-6.
- U (ultraviolet): 1.0mm UG1 + 1.0mm BG29 + 3.0mm fused silica
- B (blue): 2.0mm GG395 + 2.0mm BG39 + 1.0mm BG12
- V (visual): 3.0mm GG495 + 3.0mm BG39. Originally, the long wavelength cutoff
of the V band was set by the red response of the S1 photocathode.
- R (red): 2.0mm OG570 + 3.0mm KG3 (Kron system)
- I (infra-red): 2.0mm WG305 + 3.0mm RG9 (Kron system)
The photographic band (m_{pg}) is from photographic plates with
a IIaO response. For most stars, m_{pg}-m_{V}=B-V-0.11 (see
colors).
The near-IR bands (J, H, K, L, M, N, Q) are not named mnemonically, but
almost
alphabetically following I. A review of these bandpasses is given
Bessell and Brett (PASP 100, 1143, 1988). These bands are
set by
atmospheric transmission.
Intermediate bands
The Stromgren photometric system (Stromgren, QJRAS 4, 8, 1963;
Crawford and Mander AJ 71, 114, 1966)
uses intermediate-width bands (a few hundred Angstroms). It is very useful
for determining parameters, such as metallicities, temperatures, and absolute
magnitudes of hot stars (e.g., Napiwotski et al.
A&A 268, 653, 1993).
These filters are called u, v, b, y, and β.
Narrow bands
Narrow band filters are more specialized. These are often used for selecting
bright emission lines, or narrow regions of the continuum. Typical narrow-band
filters have 10-50 Angstrom widths.
Neutral Density Filters
Sometimes objects are too bright to observe without saturating the detector.
Instead of using a smaller
telescope, one can attenuate the light using a neutral density (ND) filter.
A perfect ND filter will be grey - that is, it will attenuate all wavelengths
equally. In practice, this is hard to achieve, so ND filters must be
calibrated individually. ND filters are generally used together with a
wavelength-sorting filter.
The amount of attenuation in astronomical ND filters is generally
measured in magnitudes.
An ND5 filter will attenuate light by 5 magnitudes, or a factor of 100.
On occasion, and NDx filter will attenuate light by a factor of
10^{x}. The ND filters on the ANDICAM camera are such. This can
be annoying, if you assume x means magnitudes, because 10^{x} is
>>2.5^{x} for x>1.
Note that ND filters for photographic or other purposes may use a different
definition of attenuation. here an ND5 filter may attenuate light by a
factor of 5, or by 5 optical depths (e^{5}, which is close to
5 mag).
The magnitude scale is defined such that log(S)=-0.4m+c,
where m is the magnitude and c is a constant.
The zero point of the magnitude scale is set by standard stars. Vega
(α Lyrae)
is the primary standard. By definition, the mean colors
of 11 Vega-like stars (spectral type A0) are zero.
For the V band, centered at
5500A, m_{V}=0 corresponds to
S=3.6x10^{-9}erg/cm^{2}/s/A (Rydgren et al.
1984, US Naval Obs. Pub. XXV, Pt. 1). Note that Matthews and Sandage (ApJ
138, 30, 1963) give 3.6X10^{-9}, and Allen in (Astrophysical
Quantities) gives 3.836X10^{-9}. See here
for a table of absolute calibrations of various astronomical bands.
Magnitudes are a logarithmic measure of flux, and so are dependent upon the
distance to the object.
The absolute magnitude is the magnitude an object would appear to have at a
distance of 10 parsecs (32.6 light years). Absolute magnitudes are indicated
by using capital M. From the inverse-square law, one can show that
m-M = 5 log D - 5,
where D is the distance in parsecs. the quantity m-M is called the
distance modulus.
The bolometric flux is the flux integrated over all wavelengths.
The bolometric magnitude is the corresponding magnitude.
The bolometric correction is the difference between the bolometric
and visual magnitudes, i.e., BC=m_{bol}-m_{V}. The
bolometric correction is non-negative.
The color index is the difference in magnitude between two bands, which is
proportional to the the ratio of the fluxes in the two bands. For example,
the B-V color index is m_{B}-m_{V}. B-V is zero for Vega
(by definition), and is about 0.61 for the Sun.
Commonly used color indices include:
- U-B: measures temperature and luminosity on hot stars, due to
sensitivity to the Balmer jump. Affected by reddening, line blanketing,
and, in active cool stars, chromospheric emission.
- B-V: proportional to temperature,
affected by reddening (E_{(B-V)} = A_{V}/R) and the
interstellar extinction.
- b-y: measures T, insensitive to [Fe/H]
- c_{1} = (u-v)-(b-v): proportional to the Balmer
discontinuity, proportional to M_{V}
- m_{1} = (v-b)-(b-y): proportional to the line
blanketing
- β: a narrow band color index proportional to the equivalent width
of H-β;
free of blanketing and reddening effects. Proportional to temperature in B, A,
and F stars.
Some temperature - color relations:
- T = 8065 - 3580 (B-V) (1.0 - 0.196 [Fe/H]); (0.3<B-V<0.63)
- θ_{eff} = 0.993 (R-I)_{J} + 0.539
- θ_{eff} = 0.245 (V-K) + 0.514
- θ_{eff} = 1.069 (b-y) + 0.483
- log(T_{eff}) = 3.9255 -0.31661x + 0.11780x^{2} - 0.049392x^{3}
- B-V = -737.243 + 603.853y - 163.455y^{2} + 14.6402y^{3}
- T = 8290 - 6200 (b-y) (1.0-0.108z) ; (0.2<b-y<0.4)
- T=11320 sqrt(β-2.311);
2.595 <β< 2.715 (F stars)
θ_{eff} = 5040/T
x=B-V
y=log(T_{eff})
z=0.2-10δm_{1}
There are also simple relations between the color and magnitude of a star
and
its angular diameter. This is the Barnes-Evans relation.
References:
Gilliland, 1985, ApJ 299, 286
Laird, 1985, ApJS, 57, 389
This image shows schematically the depth
into the atmosphere that radiation can penetrate.
We have to look through the atmosphere to see astronomical objects, and the
presence of the atmosphere affects the transmission of the light. Photons are
both absorbed and scattered from the path. The
absorption coefficient for constituent i of the atmosphere k_{i}
is
k_{i} = σ_{i}
n_{i} / r_{i} ρ_{0},
where σ is
the cross section (a function of wavelength),
n = is the number density,
r = is the fractional abundance, and
ρ_{0} is the density of air.
n, r, and ρ_{0}
are functions of height in the atmosphere.
The optical depth τ_{i}
through the atmosphere is given by the integral, from your elevation
z_{0} to
infinity, of the product of r_{i}(z), k_{i}(z),
and ρ_{0}(z).
The attenuation of light at an elevation z_{0} and a zenith
distance θ is given by
I(z_{0}) / I(∞) =
e^{-[sec(θ)
Σ τi(z0)]}
where I(∞) is the
brightness at the top of the atmosphere.
sec(θ) is known as the air mass (AM).
The atmosphere is opaque to X-rays and UV radiation shortward of about
3300 Angstroms due primarily absorption by O_{3}, but O,
O_{2}, N, N_{2}, and H_{2}O also provide significant
opacity.
In the near infra-red (1-20μm), H_{2}O and CO_{2} bands
dominate. At longer wavelengths, H_{2}O is opaque to the sub-mm
band.
You do not have to solve for the extinction every time you observe, although
you do need to observe sufficient standards to determine the zero-point
correction. The mean extinction is tabulated for many observatories, e.g.,
for the Kitt Peak National Observatory.
The atmospheric transmission is given by 10^{-EXT * AM}, where
AM is the air mass (= sec(zenith distance) for zenith distance
< 60^{o}).
Atmospheric Scattering
Scattering does 2 things, neither of which is good:
- it removes light from the beam, and
- it increases background levels by scattering light into the beam.
Rayleigh scattering: by molecules (λ>>r)
σ_{R}(λ) = 32 π^{3} (N-1)^{2}/3N^{2}λ^{4},
where n is the index of refraction and N is the molecular number density.
n-1 ~ 80 x 10^{-6} P(mb)/T(K)
Rayleigh scattering is not isotropic:
J(θ) = I_{0}σ_{R} 3/4 (1+cos^{2}θ) dω/4π is the intensity scattered into solid angle dω at an angle θ.
Aerosol scattering: due to dust and other large particles.
Derive using Mie scattering theory.
A sphere of radius a has a scattering
cross section σ=πa^{2}Q, where Q is composed of scattering and absorption components.
For
- a>>λ: Q_{scatt} = Q_{abs}=1 and σ=2πa^{2}
- a>λ: Q(λ)
is complex.
- dielectric spheres, including water droplets and silica dust grains:
Q ~ λ^{-1}
Light is refracted when it passes through a medium of varying index of
refraction. The index of refraction of air at STP is about 1.00029, while the
index of refraction of a vacuum is 1.0. The effect of refraction is to make
an object appear closer to the zenith than it really is. The trajectory of the
photon has a curvature K, and one can show that
nK = -sin a (dn/dz)
where n is the index of refraction, z is the vertical axis, and
a is the angle of incidence of the trajectory at the top of the
atmosphere. On
the assumption that the change da in the angle a is small,
one can integrate this trajectory through the atmosphere to find that
da = -tan(a) dn For a=45^{o}, the
deviation of the trajectory is about 1 minute of arc.
The index of refraction is wavelength dependent, varying from about
1.0003049 at 3200 Angstroms to 1.0002890 at 1 micron.
The night sky is not truly dark, as any observer from Long Island can tell you.
There are a number of significant sources of emission in the night sky, even at
the darkest sites on the Earth. This sky brightness adds noise to all
astronomical observations, and so one seeks to reduce the sky brightness as much
as possible. Sky brightness is measured in units of
magnitudes
per square arcsec, or
Janskys per square arcsec.
Sources of emission include
- Thermal emission.
The atmosphere is a 300K black body, and we are
looking through it. This is mostly a concern in the infra-red, because
the peak emissivity of a 300K black body is near a wavelength of 10 microns.
The following table gives typical night sky brightnesses, in magnitudes
per square arcsec and Janskys per square
arcsec, in four near-IR bands.
Band |
L |
M |
N |
Q |
microns |
3.45 |
4.8 |
10 |
20 |
mag/sq. arcsec |
8.1 |
2. |
-2.1 |
-5.8 |
Jy/sq. arcsec |
0.16 |
22.5 |
250 |
2100 |
- Airglow. This is fluorescent emission from atoms and
molecules in the Earth's atmosphere. Important airglow lines include
the He I (304Å) and H I (1216Å) Lyman-alpha lines from the
geocorona, OH and O_{2} lines in the UV and EUV,
O_{2} 5577Å and 6300Å, and O_{3} and
H_{2}O lines in the red and infra-red. Typical line intensities
range up to a few thousand Rayleighs (R;
1000R=kR) for the strongest lines. The line strength is a strong
function of solar zenith angle. Airglow is markedly reduced above a few
hundred km altitude for all but the strongest lines arising from the
Earth's geocorona, including He I (304Å) and H I (1216Å).
Observations down the Earth's shadow cone are much less strongly affected.
1 R/Å = 22 mag/sq. arcsec.
- Scattered sunlight and moonlight
from dust and aerosols in the Earth's
atmosphere. Scattered sunlight is not a problem after the end of
astronomical twilight. The amount of
scattered moonlight is a strong function of lunar phase and the distance
from the moon. The sky brightness may reach 15 mag/sq. arcsec
10^{o} from the quarter moon.
- Scattered ground illumination. This is man-made light reflected
downward from dust and aerosols in the atmosphere. It is a problem for
observatories on college campuses, or near big cities. It depends on
the local area, the time of night, and the amount of haze in the
atmosphere.
- Zodiacal light, which is sunlight scattered from
interplanetary dust in
the ecliptic plane. Out of the ecliptic,
the sky brightness at 4250Å is about 23.5 mag/sq. arcsec.
An excellent terrestrial site will have a typical sky background in the optical
of about 22 mag/sq. arcsec in the dark of the moon, looking out of the
ecliptic plane, and between airglow lines. The sky brightness observed in the
V band from the HST ranges from about
22 mag/sq. arcsec in the ecliptic to 23.3 mag/sq. arcsec near the ecliptic
poles.
m_{*} = m_{inst} + a_{0}
+ a_{1} * C + a_{2} * AM + a_{3} * AM * C
+ a_{4}(T)
m_{*}: the true stellar magnitude
m_{inst}: the instrumental magnitude (2.5 * log (count rate))
a_{0}: zeropoint correction
a_{1}: color term
a_{2}: first order extinction coefficient
a_{3}: second order extinction coefficient
a_{4}: time dependence of atmosphere
C: the color of the target
Note that the coefficients a_{1} - a_{3} may be time
dependent (if they are, the night is not really photometric, but photometricity
can be recovered). In some cases, such as very precise work, or when dealing
with objects with extreme colors, it may be necessary to incorporate higher
order terms, such as a_{5} * C^{2}
+ a_{6} * AM^{2}.