Converting Magnitudes to SED (Spectral Energy Distribution)
Bruce L. Gary, Last Updated 2014.10.18

Suppose you want to know a star's "spectral energy distribution" (SED). If it's brighter than ~ 16th magnitude it will have good quality mag's for bands BVg'r'i'JHKs, thanks to the AAVSO's APASS project (AAVSO Photometric All-Sky Survey) and the 2MASS project (2-Micron All-Sky Survey). There's an easy conversion of these magnitudes to star flux at the bandpass equivalent wavelength.

V-mag Example

This section may look complicated, but that's just because we're going through the logic of what needs to be done. In the next section you'll see how easy it really is.

Consider the star at 043049+172116 (J2000), with V-mag = 9.978. It is fainter than a zero magnitude star by the factor 10^(-0.4*9.978) = 1.020e-5. Since a zero mag star at V band has a flux of 3836 Janskys (Jy), our star of interest has a flux of 0.391 [Jy]. Since a Jy is defined to have units of 1e-23 [erg/s/cm2/Hz], the star's flux = 3.91e-24 [erg/s/cm2/Hz]. We don't care about Jy, because we're not a radio astronomer; we want flux in units of either [erg/s/cm2/micron] or [watts/m2/micron] or [watts/m2]. Let's go through the steps for each.

Note: Just to be clear, when I write [erg/s/cm2/micron] I mean [erg / (s · cm² · micron)]. Also, from now on I'm going to abbreviate the word "flux" with F, and wavelength will be given as λ.

If we multiply F [Jy] by c/λ² we'll get F [erg/s/cm2/cm], provided we use c = 2.997924e10 [cm/s].  Since this is flux per unit wavelength interval let's call it F_λ. CCD counts are proportional to the number of photo-electrons counted, and longer wavelength photons have less energy than shorter wavelength ones. Therefore there's a good reason to create a version of F that is proportional to photon flux. This is achieved simply by multiplying F_λ [erg/s/cm2/cm] by λ [cm]. This new flux is called λF_λ[erg/s/cm2]. If you prefer λF_λ[watts/m2], as I do, then multiply by 1e-3.  Returning to the example, above, with F = 3.91e-24 [erg/s/Hz], and λ = 550 [nm] = 5.50e-5 [cm], we can calculate F_λ = 3.91e-24 [erg/s/Hz] * {2.998e10 / (5.50e-5)^2} = 3.88e-5 [erg/s/cm2]. Also, λF_λ = 2.13e-12 [W/m2] = 2.13e-12 [watt/m2].

Generalizing the V-mag Example

Mathematically, all we did in the previous example is:  λF_λ [W/m2] = 11.50e-9 × (10^{-0.4 V}) / λ [micron].  In other words, the general equation is a constant (associated with the filter band in question) times a magnitude ratio divided by wavelength in microns. To simplify further, let's define a "SED Constant" that includes the wavelength, so that:

        λF_λ [W/m²] = SED Constant × 10^{-0.4 Mag}

which allows us to present the following table listing the SED Constant:

Table 1. The last column is used to calculate star flux according to the equation λF_λ [W/m2]= SED Constant × 10^{-0.4 MAG}, where MAG is star magnitude. "Jansky Constant" is given merely for anyone wanting to know the basis for deriving the SED Constant. The g'r'i'z' SED  Constants are my empirical values.

Getting APASS & 2MASS Mag's

Creating the SED

Here's an example of a SED for the star used in the above example.

When the star is fit well by a blackbody (BB) function it is a simple matter of representing the star's flux at any desired wavelength (within the wavelength range where the fit is good). This is useful for creating a SED for target stars not in the UCAC4 catalog, and especially for asteroids. For example, using a spectrograph in an attempt to create an asteroid's SED the procedure can be to 1) measure the spectrum for a nearby star, 2) measure the spectrum for the asteroid, 3) create a SED for the nearby star using BVg'r'i'JHK magnitudes (as demonstrated above), 4) use the BB fn to calculate a flux spectrum for the nearby star at the resolution of the spectrograph, and 5) use the ratio of the asteroid's measured spectrum to the nearby star's spectrum, multiplied by the BB fn of the previous step, to arrive at the asteroid's flux spectrum. If the goal is to determine the asteroid's geometric albedo, for example, then the asteroid's flux spectrum can be divided by the sun's known flux spectrum, etc. An illustration of this, using a transmission grating (SA100) is shown at this link: http://brucegary.net/Albedo/.

Another Example

The Landolt stars have excellent quality BVRcIc magnitudes, and the Smith et al (2000) SDSS stars have excellent g'r'i'z' magnitudes. The latter catalog is a subset of the former, so it includes all of the above magnitudes. Here's an example for Star #24 in my list of Smith et al stars.

Figure 2. SED for a star with Landolt and SDSS magnitudes (as well as 2MASS mag's).

Figure 3. Same SED for the visible region.

BGary master list of web sites:  http://brucegary.net/MASTER.html

This site was opened 2014.09.06. Nothing on this web page is copyrighted.   BGary web sites