Light, magnitudes, and the signal equation

(Entire section in one PDF file).

Light

• astronomy - learn about objects outside of our atmosphere. need information from them to do so: light!

• light, by quantum mechanics, is photons, has characteristics of both waves and particles. Wavelength/frequency corresponds to energy:
  E = hν = $${h c \over \lambda}$$

• electromagnetic spectrum: gamma rays - X rays - UV - optical - IR - mm - radio. Different units often used for wavelength in different parts of spectrum: 1Å = 1×10^-10 m (used in UV, optical), 1nm = 1×10^-9 m (used in UV, optical), 1μ = 1×10^-6 m (used in IR), 1mm = 1×10^-3 m

  Numerical wavelengths of different parts of spectrum (roughly, there is no established strict vocabulary!): far-UV ( 0.01 - 0.1μ, 100-1000 Å), near-UV ( .1 - 0.35μ, 1000-3500 Å), optical ( 0.35 - 1μ, 3500-10000 Å), near-IR (1 - 10μ), mid-IR ( 10 - 100μ), far-IR ( 100 - 1000μ). Of course, some people used frequency instead of wavelength! And others, especially for high energy radiation, use energy!

• We can describe the amount of light an object emits or that we receive by three fundamental quantities: intensity or surface brightness, I, flux F, or luminosity, L.

• The surface brightness is defined as the amount of energy received in a unit surface element per unit time per unit frequency (or wavelength) from a unit solid angle in the direction ( θ, φ), where θ is the angle away from the normal to the surface element, and φ the azimuthal angle. The solid angle is related to the physical size of an object and its distance: dΩ = dA/d². Note units often used for solid angle: sterradian, square degree, square arcsec.

  The flux is the amount of energy passing through a unit surface element in all directions, defined by

  F_ν = $$\int$$I_νcosθdΩ

  where dΩ is the solid angle element, and the integration is over the entire solid angle. Usually, our detectors are pointed such that the light is received perpendicular to the collecting area and the angle subtended by an object is very small, so the cosθ term is well approximated by unity.

  The luminosity is the intrinsic energy emitted by the source per second. For an isotropically emitting source,

  L = 4πd²F

  where d = distance to source.

• What do we measure for sources? For resolved sources we can directly measure their surface brightness (intensity) distribution on the sky, usually over some bandpass or wavelength interval; for unresolved sources, we measure the flux. We can only calculate the luminosity to a source if we know the distance.

• There is an additional property of light that we haven't discussed: polarization. As a transverse wave, the electromagnetic field associated with a photon is oriented in a particular direction. For many sources of emission, all orientations are produced, and the result is unpolarized light. However, for some emission mechanisms, light can be polarized, with a fraction of it all oriented in the same plane (linear polarized) or with a plane that rotates as it propagates (circularly polarized). In particular, polarization can arise from emission that is reflected and also from emission in regions of magnetic fields. Generally, the polarization is characterezed by the Stokes paramteres, I, Q, U, and V, which give the polarization intensity, two components of linear polarization, and circular polarization.

• Note, there are variations in terminology, especially between disciplines (e.g. astronomy and engineering) and you should make sure you understand what is adopted by someone presenting data. While astronomers are reasonably consistent in the use of these terms, you may run into things like radiance( surface brightness), irradiance (flux), radiant flux (luminosity), spectral intensity, and others, which are all related to intensity or flux.

• Amount of light emitted is a function of wavelength, so we actually are often interested in estimates of the monochromatic flux/intensity/luminosity, sometimes known as flux/intensity/luminosity density = flux / unit wavelength (or unit frequency), also sometimes known as specific flux/intensity/luminosity.

  F_ν : flux per unit frequency. F_λ : flux per unit wavelength.

  $$\int_{{nu1}}^{{nu2}}$$F_νdν = - $$\int_{{lambda1}}^{{lambda2}}$$F_λdλ
  F_ν = - F_λdλ/dν
  = F_λ$${c \over \nu^2}$$ = F_λ$${\lambda^2\over c}$$

  Similarly, intensity and luminosity can be given per unit wavelength (or frequency). Note that a constant F_λ implies a non-constant F_ν and vice versa!

  Integral of flux/brightness over all wavelengths/frequencies gives the bolometric flux/brightness.

• Units: astronomers often (not always) work in CGS units, although, as discussed below, they most often work in a dimensionless unit ... magnitudes.
  • You may also run into the Jansky, a flux density unit corresponding to 10^-26W/m²/Hz

• In practice, with most detectors, we measure photon flux (photon counting devices), rather than energy flux (bolometers). The photon flux is given by
  photonflux = flux/energyperphoton
  = $$\int$$F_λ$${\lambda \over h c}$$dλ

• Terminology of measurements:
  • photometry (broad-band flux measurement),
  • spectroscopy (relative measurement of fluxes at different wavelengths),
  • spectrophotometry (absolute measurement of fluxes at different wavelengths).
  • astrometry: concerned with positions of observed flux;
  • morphology: intensity as a function of position; often, absolute measurements are unimportant

Magnitudes and photometric systems

In astronomy, however, magnitude units are often used instead of measuring the basic quantities in energy or photon flux. Magnitudes are a dimensionless quantities, and are related to flux (same holds for surface brightness or luminosity) by:

m = - 2.5 log$${F\over F_0}$$

or

m = - 2.5 log F + 2.5 log F₀

where the coefficient of proportionality, F₀, depends on the definition of photometric system; the quantity -2.5 log F₀ may be referred to as the photometric system zeropoint. This defining equation is sometimes referred to as the Pogson equation, after Pogson (1856). Inverting, one gets:

F = F₀10^-0.4m

Note that since magnitudes are logarithmic, the difference between magnitudes corresponds to a ratio of fluxes; ratios of magnitudes are generally unphysical! If one is just doing relative measurements of brightness between objects, this can be done without knowledge of F₀ (or, equivalently, the system zeropoint); objects that differ in brightness by ΔM mag have the same ratio of brightness rgardless of what photometric system they are in:

m₁ - m₂ = - 2.5 log$${F_1 \over F_2}$$

$${F_1 \over F_2}$$ = 10^{-0.4(m₁-m₂)}

The photometric system definitions and zeropoints are only needed when converting between calibrated magnitudes and fluxes. However, the utility of a system when doing astrophysics generally requires an understanding of the actual fluxes.

Luminosities are represented as absolute magnitudes, i.e., the magnitude a star would have if it were at a distance of 10 parsec; as before, you need a distance to get a luminosity. The inverse square law expressed in magnitudes leads to the distance modulus:

m₀ - M = 5 log d - 5

(derive it!), where m₀ is the apparent magnitude corrected for interstellar extinction: m₀ = m - A.

Just as fluxes can be represented in magnitude units, flux densities can be specified by monochromatic magnitudes:

F_λ = F₀(λ)10^-0.4m(λ)

although spectra are more often given in flux units than in magnitude units. Note that it is possible that F₀ is a function of wavelength!

There are three main types of magnitude systems in use in astronomy. We start by describing the two simpler ones:the STMAG and the ABNU mag system. In these simple system, the reference flux is just a constant value in F_λ or F_ν. However, these are not always the most widely used systems in astronomy, because no natural source exists with a flat spectrum.

In the STMAG system, F_{0, λ} = 3.63E - 9ergs/cm²/s/Å, which is the flux of Vega at 5500Å; hence a star of Vega's brightness at 5500Å is defined to have m=0. Alternatively, we can write

m_STMAG = - 2.5 log F_λ - 21.1

(for F_λ in cgs units).

In the ABNU system, things are defined for F_ν instead of F_λ, and we have

F_{0, ν} = 3.63×10^-20erg/cm²/s/Hz10^-0.4m_ν

or

m_ABNU = - 2.5 log F_ν - 48.6

(for F_ν in cgs units). Again, the constant comes from the flux of Vega.

Usually, when using magnitudes, people are talking about flux integrated over a spectral bandpass. In this case, F and F₀ refer to fluxes integrated over the bandpass. The STMAG and ABMAG integrated systems are defined relative to sources of constant F_λ and F_ν systems, respectively.

m_STMAG = - 2.5 log$${\int F_\lambda \lambda d\lambda\over \int 3.63\times 10^{-9} \lambda d\lambda}$$

(the factor of λ comes in for photon counting detectors).

m_ABNU = - 2.5 log$${\int (F_\nu/\nu) d\nu\over \int (3.63\times 10^{-20}/\nu) d\nu}$$

(where the units are implicitly cgs with these numerical fluxes for Vega).

Note that these systems differ by more than a constant, because one is defined by units of F_λ and the other by F_ν, so the difference between the systems is a function of wavelength. They are defined to be the same at 5500Å. (Question: what's the relation between m_STMAG and m_ABNU?)

Note also that, using magnitudes, the measured magnitude is nearly independent of bandpass width (a broader bandpass does not imply a brighter (smaller) magnitude), which is not the case for fluxes!

The standard UBVRI broadband photometric system, as well as several other magnitude systems, however, are not defined for a constant F_λ or F_ν spectrum; rather, they are defined relative to the spectrum of an A0V star. Most systems are defined (or at least were originally) to have the magnitude of Vega be zero in all bandpasses (VEGAMAGS); if you ever get into this in detail, note that this is not exactly true for the UBVRI system.

For the broadband UBVRI system, we have

m_UBVRI $$\approx$$ -2.5 log$${\int_{UBVRI} F_\lambda(object) \lambda d\lambda\over \int_{UBVRI} F_\lambda(Vega) \lambda d\lambda}$$

(as above, the factor of λ comes in for photon counting detectors).

Here is a plot to demonstrate the difference between the different systems.

Why do the different systems exist? While it seems that STMAG and ABNU systems are more straightforward, in practice it is difficult to measure absolute fluxes, and much easier to measure relative fluxes between objects. Hence, historically observations were tied to observations of Vega (or to stars which themselves were tied to Vega), so VEGAMAGs made sense, and the issue of determining physical fluxes boiled down to measuring the physical flux of Vega. Today, in some cases, it may be more accurate to measure the absolute throughput of an instrumental system, and using STMAG or ABNU makes more sense.

Colors

Working in magnitudes, the difference in magnitudes between different bandpasses (called the color index, or simply, color) is related to the flux ratio between the bandpasses, i.e., the color. In the UBVRI system, the difference between magnitudes gives the ratio of the fluxes in different bandpasses relative to the ratio of the fluxes of an A0V star in the different bandpasses (for VEGAMAG). Note the typical colors of astronomical objects – which are different for the different photometric systems!

Which is closer to the UBVRI system, STMAG or ABNU?

What would typical colors be in an STMAG or ABNU system?

Magnitude-flux conversion

How would one go about converting Vega-based magnitudes to fluxes? Roughly, just look up the flux of Vega at the center of the passband ( e.g., here (from Bessell et al 1998 or here (see references within), or here; note, however, if the spectrum of the object differs from that of Vega, this won't be perfectly accurate (see, e.g. discussion of WISE photometry) Given UBVRI magnitudes of an object in the desired band, filter profiles (e.g. Bessell 1990, PASP 102,1181), and absolute spectrophotometry of Vega (e.g., Bohlin & Gilliland 2004, AJ 127, 3508, one can determine the flux.

If one wanted to estimate the flux of some object in arbitrary bandpass given just the V magnitude of an object (a common situation used when trying to predict exposures times, see below), this can be done if an estimate of the spectral energy distribution (SED) can be made (e.g., from the spectral type, or more generally, the stellar parameters T_eff, log g, and metallicity). Given the filter profiles, one can compute the integral of the SED over the V bandpass, determine the scaling by comparing with the integral of the Vega spectrum over the same bandpass, then use the normalized SED to compute the flux in any desired bandpass. Some possibly useful references for SEDs are: Pickles atlas, MILES library, Bruzual, Persson, Gunn, & Stryker; Hunter, Christian, & Jacoby; Kurucz).

Things are certainly simpler in the ABNU or STMAG system, and there has been some movement in this direction: the STScI gives STMAG calibrations for HST instruments, and the SDSS photometric system is close to an ABNU system.

Note, however, that even when the systems are conceptually well defined, determining the absolute calibration of any photometric system is very difficult in reality, and determining absolute fluxes to the 1% level is very challenging.

As a separate note on magnitudes themselves, note that some people, in particular, the SDSS imaging survey, have adopted a modified type of magnitudes, called asinh magnitudes, which behave like normal (also known as Pogson) magnitude for brighter objects, but have different behavior for very faint objects (near the detection threshold); see Lupton, Gunn, & Szalay 1999 AJ 118, 1406 for details.

Observed fluxes, the signal equation, and photometry

What if you are measuring flux with an actual instrument, i.e. counting photons? The intrinsic photon flux from the source is not trivial to determine from the observed photon flux, i.e., the number of photons that you count. The observed flux depends on the area of your photon collector (telescope), photon losses and gains from the Earth's atmosphere (which changes with conditions), and the efficiency of your collection/detection apparatus (which can change with time). Generally, the astronomical signal (which might be a flux or a surface brightness, depending on whether the object is resolved) can be written

S = Tt$$\int$$$${F_\lambda\over{hc\over\lambda}}$$a_λtel_λinst_λfilt_λdet_λdλ≡TtS'

where S is the observed photon flux (the signal), T is the telescope collecting area, t is the integration time, a_λ is the atmospheric transmission (more later) and the other terms refer to the efficiency of various components of the system (telescope, instrument, filter, detector). S' is an observed flux rate, i.e. with all of the real details of the observing system included. I refer to this as the signal equation.

Usually, however, one doesn't use this information to go backward from S to F_λ because it is very difficult to measure all of the terms precisely, and some of them (e.g. a, and perhaps some of the system efficiencies) are time-variable; a is also spatially variable.

While the signal equation isn't usually used for calibration, it is very commonly used for computing the approximate number of photons you will receive from a given source in a given amount of time for a given observational setup. This number is critical to know in order to estimate your expected errors and exposure times in observing proposals, observing runs, etc. Understanding errors in absolutely critical in all sciences, and maybe even more so in astronomy, where objects are faint, photons are scarce, and errors are not at all insignificant. The signal equation provides the basis for exposure time calculator (ETC) programs, because it gives an expectation of the number of photons that will be received by a given instrument as a function of exposure time. As we will see shortly, this provides the information we need to calculate the uncertainty in the measurement as a function of exposure time.

Photometry

So if we don't use the signal equation for calibration, how do we go about determining calibrated brightnesses from measurements? To do this, most observations are performed differentially to a set of other stars of known brightness. If one or more stars of known brightness are observed in the same observation, then the atmospheric term is (approximately) the same for all stars; this is known as differential photometry. From the photon flux of the object with known brightness, one can calculate an instrumental magnitude:

m = - 2.5 log(S/t)

and then determine the zeropoint that needs to be added to give the calibrated magnitude (M, make sure you recognize that this is still an apparent magnitude!):

M = m + z

Note that the zeropoint gives a measure of the system sensitivity: it is the magnitude of an object which produces 1 count/s, so a larger zeropoint indicates a more sensitive system (i.e., from larger aperture, throughput, etc.); alternatively, one can calculate an ``effective area" for an exposure. The normalization by the exposure time in the instrumental magnitude to get counts/sec is not strictly necessary, but it is useful if you are using the zeropoint from one exposure to calibrate another exposure of a different exposure time.

Note that in the real world, one has to also consider sensitivity differences (e.g., slightly different filter profiles) between a given experimental setup and the setup used to measure the reference brightnesses. If the experimental system differs in response details to the standard system, the zeropoint will be different for objects with different spectral energy distributions. Usually, at attempt is made to calibrate this using so-called tranformation coefficients and parametrizing the SED differences by the color of the objects. The relation between the instrumental magnitude and the standard magnitude is given by:

M = m + t(color) + z

where capital letters are the magnitude on the standard system, z is the zeropoint, and t is the transformation coefficent. There is a separate such relation for each filter in which observations are made.

The color is generally parameterized by the ratio of the flux at two different wavelengths, or, in magnitudes, the difference between the magnitudes. The two wavelengths should be measured near in wavelength to the wavelength of the filter being corrected; generally, one uses the bandpass being corrected as one of the wavelenghts and an adjacent bandpass as the other. For example, when correcting V magnitudes, people usually use B - V, V - R, or V - I for the color term, e.g.:

V = m_V + t_V(B - V) + z_V

Clearly, to do this solution, you need more than one standard star, since there are two unknowns (t_V and z_V), and to get a meaningful estimate of t_V, you want the standard stars to cover as wide a range of color as possible. While you can solve for the coefficients with two stars, one generally would like to have more than this, and solve for the coefficients using, e.g., least squares.

There are two ways to define the color, either in terms of the observational system or in terms of the standard system. The latter is slightly preferred for using least-squares (small errors on the independent variable), and also because it allows observations from different nights to be combined. Note that this formulation does not require you to know the colors of your objects a priori, it's just algebra to figure them out as long as you have observations in both filters, e.g., once you have the transformation coefficients and the zeropoints for two filters, you can solve:

B = m_B + t_B(B - V) + z_B

V = m_V + t_V(B - V) + z_V

for both B and V given m_B, m_V, t_B, t_V, z_B, and z_V.

The use of these first-order transformation coefficients is accurate as long as your filter system does not differ much from the standard system, and additionally, that the spectrum of your program objects does not differ significantly from the spectrum of the standard objects. The more these conditions are not met, the less accurate the results. Some additional accuracy in the case of differing systems can be achieved by using higher order transformation coefficients. However, even in this case, it is always important to remember that if the spectrum of the program object differs significantly from the standards, derived fluxes can be significantly in error.

Certainly, you get to a point when the response of one system is so different than the response of another system that no transformation can be determined. In this case, you have two different photometric systems. In fact, there are several different photometric systems at use in astronomy today, and each has advantages and disadvantages.

If there are no stars of known brightness in the same observation, then calibration must be done against stars in other observations. This then requires that the different effects of the Earth's atmosphere in different locations in the sky be accounted for. This is known as all-sky, or absolute, photometry. To do this requires that the sky is ``well-behaved", i.e. one can accurately predict the atmospheric throughput as a function of position. This requires that there be no clouds, i.e. photometric weather. Differential photometry can be done in non-photometric weather, hence it is much simpler! Of course, it is always possible to obtain differential photometry and then go back later and obtain absolute photometry of the reference stars. We will discuss later how to incorporate the effects of the Earth's atmosphere. However, all-sky photometry is becoming less and less common as catalogs of well calibrated stars are becoming available across the entire sky (e.g., SDSS or PanSTARRS).

Of course, at some point, someone needs to figure out what the fluxes of the calibrating stars really are, and this requires understanding all of the terms in the signal equation. It is challenging, and often, absolute calibration of a system is uncertain to a couple of percent!

It is also common to stop with differential photometry, even if there are no stars of known brightness in your field, if you are studying variable objects, i.e. where you are just interested in the change in brightness of an object, not the absolute flux level. In this case, one only has to reference the brightness of the target object relative some other object (or ensemble of objects) in the field that are non-variable. One has to be careful that the reference object is itself not a variable, and this becomes more challenging if you are trying to measure small variations in brightness.