(Entire section in one PDF file).
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!
The flux is the amount of energy passing through a unit surface element in all directions, defined by
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,
where d = distance to source.
Fν : flux per unit frequency. Fλ : flux per unit wavelength.
Similarly, intensity and luminosity can be given per unit wavelength (or frequency). Note that a constant Fλ implies a non-constant Fν and vice versa!
Integral of flux/brightness over all wavelengths/frequencies gives the bolometric flux/brightness.
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:
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:
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:
Just as fluxes can be represented in magnitude units, flux densities can be specified by monochromatic magnitudes:
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 Fλ or Fν. 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, F0, 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
In the ABNU system, things are defined for Fν instead of Fλ, and we have
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 Fλ and Fν systems, respectively.
Note that these systems differ by more than a constant, because one is defined by units of Fλ and the other by Fν, 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 mSTMAG and mABNU?)
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 Fλ or Fν 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
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.
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?
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 Teff, 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.
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
Usually, however, one doesn't use this information to go backward from S to Fλ 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.
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:
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:
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.:
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:
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.