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 Fλ. 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 Fλ [erg/s/cm2/cm] by λ [cm]. This new flux is called λFλ[erg/s/cm2]. If you prefer λFλ[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 Fλ = 3.91e-24 [erg/s/Hz] * {2.998e10 / (5.50e-5)^2} = 3.88e-5 [erg/s/cm2]. Also, λFλ = 2.13e-12 [W/m2] = 2.13e-12 [watt/m2].

Generalizing the V-mag Example

Mathematically, all we did in the previous example is:  λFλ [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:

λFλ [W/m2] = SED Constant × 10-0.4 Mag

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

 Band Effective Wavelength [micron] Jansky Constant [erg/s/cm2/Hz] SED Constant [W/m2] B 0.433 4267 2.954E-8 V 0.550 3836 2.091E-8 Rc 0.640 3080 1.443E-8 Ic 0.790 2550 0.968E-8 g' 0.481 2.48E-8 r' 0.670 1.75E-8 i' 0.790 1.40E-8 z' 0.910 1.22E-8 J 1.235 1594 3.869E-9 H 1.662 1024 1.847E-9 Ks 2.159 667 0.926E-9

Table 1. The last column is used to calculate star flux according to the equation λFλ [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

My favorite planetarium program for obtaining magnitudes is C2A (http://www.astrosurf.com/c2a/english/). This is a free program, and it supports the UCAC4 database if you have it on your computer. TheSky6 and TheSkyX also support UCAC4, but C2A is easier to use.

The UCAC4 star catalog (released in 2012 by USNO) includes APASS (Data Release 6) BVg'r'i' magnitudes for 51 million stars. It includes JHKs magnitudes for 110 million stars from the 2MASS catalog. You can download the UCAC4 catalog from this web site: http://www.ap-i.net/skychart/en/news/ucac4_catalog

Many planetarium programs include the 2MASS magnitudes but don't have the APASS magnitudes. An alternative way to get APASS magnitudes is to querry the AAVSO web site: http://www.aavso.org/download-apass-data  That site give info for a RA/DE circular region, which can be as small as 0.01 degree radius in order to get mag's for just the star of interest (usually).

Creating the SED

Here's an example of a SED for the star used in the above example. Figure 1. SED for a 10th magnitude star with a blackbody fit (Teff = 3750 K). The set of green bars at 0.55 micron show flux levels for integer V-mag's. (If you prefer flux units of [erg/s/cm2] simply multiply the Y-axis values by 1e-3.)

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.

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