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/cm2/s/A.

Bandpasses

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:

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 RC, IC. 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.

The photographic band (mpg) is from photographic plates with a IIaO response. For most stars, mpg-mV=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 10x. The ND filters on the ANDICAM camera are such. This can be annoying, if you assume x means magnitudes, because 10x is >>2.5x 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 (e5, which is close to 5 mag).


Magnitudes

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, mV=0 corresponds to S=3.6x10-9erg/cm2/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.


Absolute Magnitudes

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.

Bolometric Fluxes and Magnitudes

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=mbol-mV. The bolometric correction is non-negative.

Colors

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 mB-mV. B-V is zero for Vega (by definition), and is about 0.61 for the Sun.

Commonly used color indices include:

Some temperature - color relations:

θeff = 5040/T
x=B-V
y=log(Teff)
z=0.2-10δm1

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


Atmospheric Extinction


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 ki is

ki = σi ni / ri ρ0,
where σ is the cross section (a function of wavelength),
n = is the number density,
r = is the fractional abundance, and
ρ0 is the density of air.
n, r, and ρ0 are functions of height in the atmosphere.

The optical depth τi through the atmosphere is given by the integral, from your elevation z0 to infinity, of the product of ri(z), ki(z), and ρ0(z).

The attenuation of light at an elevation z0 and a zenith distance θ is given by

I(z0) / 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 O3, but O, O2, N, N2, and H2O also provide significant opacity.

In the near infra-red (1-20μm), H2O and CO2 bands dominate. At longer wavelengths, H2O 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 < 60o).


Atmospheric Scattering

Rayleigh scattering: by molecules (λ>>r)
σR(λ) = 32 π3 (N-1)2/3N2λ4,
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) = I0σR 3/4 (1+cos2θ) 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=πa2Q, where Q is composed of scattering and absorption components.


Atmospheric Refraction

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=45o, 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.


Sky Brightness

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

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.

Absolute Photometry

m* = minst + a0 + a1 * C + a2 * AM + a3 * AM * C + a4(T)
m*: the true stellar magnitude
minst: the instrumental magnitude (2.5 * log (count rate))
a0: zeropoint correction
a1: color term
a2: first order extinction coefficient
a3: second order extinction coefficient
a4: time dependence of atmosphere
C: the color of the target
Note that the coefficients a1 - a3 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 a5 * C2 + a6 * AM2.