# Filters

I found this page which was a helpful expansion on David's email of Feb. 11. Note however that the table for Photon Flux has different central wavelengths for the AB filters than David did (the data for the Johnson filters all match). --Childers 18:19, 19 February 2012 (PST)

## Convert Johnson Magnitude to Janskys

Solving the magnitude formula for the flux yields:

flux = 10^(-0.4*magnitude) * zero_point_flux ~ (0.3981)^magnitude * zero_point_flux

test $\displaystyle f = 10^{-0.4m} \times f_0 \sim (0.3981)^{m} \times f_0$ $\displaystyle B_{\lambda} = \left(\frac{2hc^2/\lambda^5}{\exp(hc/\lambda kT)-1)}\right)$

$\displaystyle f = 10^{(-0.4m)} f_0 \approx (0.3981)^m f_0$ test

Since the magnitude is a dimensionless quantity, the units for the flux will be identical to the units for the zero_point_flux. In the table below, use the right-most column.

## Convert AB Magnitude to Janskys

Because the AB magnitude system is defined by a fixed zero-point flux instead of a reference object, any AB magnitude can be converted to janskys with this formula:

flux = (0.3981)^magnitude * 3631 Jy

$\displaystyle \phi = (0.3981)^m \times 3631 \mathrm{Jy}$

## David's Original E-Mail

Dear Folks,

As you know magnitudes are a logarithmic unit that runs backwards (i.e., brighter sources have smaller values). The magnitude scale is defined such that

magnitude = -2.5*log10(flux/zero_point_flux)

$\displaystyle m=-2.5\log{\frac{\phi}{\phi_0}}$

where flux is the flux of the star and the zero_point_flux is the flux of a reference star DEFINED to have a magnitude = 0.0

Magnitudes are useful handles on comparing the brightness of two or more sources within the same filter; i.e., source A has a magnitude of V=10 and source B has a magnitude of V=5 therefore source A is 100 times fainter). But in order to compare brightnesses of the SAME object at different filters or to compare other physical units (like the total luminosity), the magnitude scale needs to be converted to flux. For example, Vega is DEFINED to have a Johnson V=0 mag and a Johnson K = 0 mag; however, Vega is 5 times brighter in the optical (V) than it is in the infrared (K).

And just to make things more complicated, there are actually two different kinds of magnitudes. There is the Johnson magnitude system (the one you are most familiar with). In this system, Vega is the standard where all magnitudes are compared to Vega which is DEFINED to have a magnitude = 0 in every filter. These filters typically include (and capitalization matters here) UBVRIJHK. In this system, the zero_point_flux for each filter changes, because the brightness of Vega changes with wavelength.

Filter Central wave (microns) Width wave (microns) zero point flux (Jy)
U 0.36 0.15 1810
B 0.44 0.22 4260
V 0.55 0.16 3640
R 0.64 0.23 3080
I 0.79 0.19 2550
J 1.26 0.16 1600
H 1.60 0.23 1080
K 2.22 0.23 670
Ks 2.15 0.31 666.7

(1 Jy = 1 Jansky = 10^-26 W Hz^-1 m^-2)

There is another type of magnitude system called the AB magnitude system where Vega is not the standard, but rather every filter is DEFINED to have the same zero_point_flux of 3631 Jy for all filters in the AB system. The ugriz filters in the Kepler catalog are in the Sloan Digitized Sky Survey (SDSS) system which is an AB magnitude system.

Central wavelengths of these filters are ...

u g r i z
0.355 0.469 0.617 0.748 0.893

Finally, just to make things even more confusing, there are various versions of the filters (e.g., Johnson R vs Cousins R) that are similar but slightly different in wavelength or zero point flux or both. For our purposes, I would not worry so much about that.

One more finally, in the KIC, you will see two filter D51 and GREDMAG - those are non-standard filters and finding conversions for those is difficult. You can collect them but I will need some time to think about how to use them.

Also one more finally, I used the term flux. Actually, these conversions are from magnitude to FLUX DENSITY. A flux density is an amount of energy per time per area per unit wavelength or frequency (hence the per Hz in the units of Jansky) whereas a flux is an amount of energy per unit time per area over all wavelengths. The terminology is a detail that astronomers often use interchangeably - and often incorrectly.

$\displaystyle B_{\lambda} = \left(\frac{2hc^2/\lambda^5}{\exp(hc/\lambda kT)-1)}\right)$