Photon Flux Density vs. Energy Flux Density

One of the subtleties of photometry is the difference between magnitudes and colors calculated using energy flux densities (EFD) and photon flux densities (PFD).

The complication arises since the photometry presented by many surveys is calculated using PFD but spectra (specifically the synthetic variety) is given as EFD. The difference is small but measurable so let's do it right.

The following is the process I used to remedy the situation by switching my models to PFD so they could be directly compared to the photometry from the surveys. Thanks to Mike Cushing for the guidance.

Filter Zero Points

Before we can calculate the magnitudes, we need filter zero points calculated from PFD. To do this, I started with a spectrum of Vega in units of [erg s-1 cm-2 A-1] snatched from STSci.

Then the zero point flux density in [photons s-1 cm-2 A-1] is:

Where is the given energy flux density in [erg s-1 cm-2 A-1] of Vega, is the photon flux density in [photons s-1 cm-2 A-1], and is the scalar filter throughput.

Since I'm starting with a spectrum of Vega in EFD units, I need to multiply by to convert it to PFD units.

In Python, this looks like:

def zp_flux(band):
    from scipy import trapz, interp, log10
    (wave, flux), filt, h, c = vega(), get_filters()[band], 6.6260755E-27 # [erg*s], 2.998E14 # [um/s]
    I = interp(wave, filt['wav'], filt['rsr'], left=0, right=0)
    return trapz(I*flux*wave/(h*c), x=wave)/trapz(I, x=wave))

Calculating Magnitudes

Now that we have the filter zero points, we can calculate the magnitudes using:

Where is the apparent magnitude and is the flux from our source given similarly by:

Since the synthetic spectra I'm using are given in EFD units, I need to multiply by to convert it to PFD units just as I did with my spectrum of Vega.

In Python the magnitudes are obtained the same way as above but we use the source spectrum in [erg s-1 cm-2 A-1] instead of Vega. Then the magnitude is just:

mag = -2.5*log10(source_flux(band)/zp_flux(band))

Below is an image that shows the discrepancy between using EFD and PFD to calculate colors for comparison with survey photometry.

The circles are colors calculated from synthetic spectra of low surface gravity (large circles) to high surface gravity (small circles). The grey lines are iso-temperature contours. The jumping shows the different results using PFD and EFD. The stationary blue stars, green squares and red triangles are catalog photometric points calculated from PFD.

The circles are colors calculated from synthetic spectra of low surface gravity (large circles) to high surface gravity (small circles). The grey lines are iso-temperature contours. The jumping shows the different results using PFD and EFD. The stationary blue stars, green squares and red triangles are catalog photometric points calculated from PFD.

Other Considerations

The discrepancy I get between the same color calculated from PFD and EFD though is as much as 0.244 mags (in r-W3 at 1050K), which seems excessive. The magnitude calculation reduces to:

Since the filter profile is interpolated with the spectrum before integration, I thought the discrepancy must be due only to the difference in resolution between the synthetic and Vega spectra. In other words, I have to make sure the wavelength arrays for Vega and the source are identical so the trapezoidal sums have the same width bins.

This reduces the discrepancy in r-W3 at 1050K from -0.244 mags to -0.067 mags, which is better. However, the discrepancy in H-[3.6] goes from 0.071 mags to -0.078 mags.

To Recapitulate

In summary, I had a spectrum of Vega and some synthetic spectra all in energy flux density units of [erg s-1 cm-2 A-1] and some photometric points from the survey catalogs calculated from photon flux density units of [photons s-1 cm-2 A-1].

In order to compare apples to apples, I first converted my spectra to PFD by multiplying by at each wavelength point before integrating to calculate my zero points and magnitudes.

Colors Diagnostic of Surface Gravity

The goal here is to find a prescription of colors diagnostic of brown dwarf surface gravity. Since early optical as well as far infrared spectra and photometry are uncommon, the bands of interest should only include i and z from SDSS; J, H and Ks from 2MASS; and W1, W2 and W3 (but not W4 with only 10 percent detection) from WISE.

In order to find said prescriptions, I used the BT-Settl models (at solar metallicity ranging from 1000 - 3000 K in effective temperature and 3.0 - 5.5 dex in log surface gravity) to produce a suite of color-color and color-parameter plots.

One method I employed was to choose one effective temperature (in this case 2500K) and anchor the colors in one band that doesn't vary much between high and low surface gravity, e.g. z-band. Then I chose the other two bands by one that was more luminous at low gravity and one that was more luminous at high gravity, e.g. W2- and J-band respectively.

BT-Settl model spectra at 2500K

Then the color-color plot of these bands looks like:

In this plot of z-J vs. z-W2 the smallest circles are objects with high surface gravity and the largest have low surface gravity (log(g) = 5.5 to 3.5 respectively). The light grey lines are iso-temperature contours.

In this plot of z-J vs. z-W2 the smallest circles are objects with high surface gravity and the largest have low surface gravity (log(g) = 5.5 to 3.5 respectively). The light grey lines are iso-temperature contours.

In this particular case, there is little-to-no dispersion in z-J for Teff = 2500K (d = 0.009) and an appreciable dispersion in z-W2 for that same Teff (d = 0.32). Notice the tight vertical grouping (z-J) and dispersed horizontal grouping (z-W2) for the model objects of Teff = 2500K and varying log(g) in the red rectangle on the color-color plot above.

Double-checking with the color-Teff plots, we can see that the dispersion in z-J in the plot on the left is tiny and the horizontal offset in the color-color plot is due to the 0.32 magnitude dispersion in z-W2 on the right below.

Of course this is just a different way of looking at the same thing, but I might be able to find colors that are reliable indicators of gravity (and thus age) if I can find a bunch of these examples where the flux in the secondary and tertiary bands are flipped.

Of note is the fact that at this temperature in this color-color plot the points are also isolated, i.e. there are no degeneracies with objects of any other temperature. That means that if I find an object with a z-J = 1.65 or so, I know that it has an effective temperature of about 2500K. Then I can determine its age by seeing if its z-W2 color is closer to 3.3 (young) or 2.9 (old).

This of course does not work for all temperatures, as shown in the red circle in the color-color plot above. This demonstrates a degeneracy among hotter young objects (Teff = 3000K, log(g) = 3.5) and cooler old objects (Teff = 2800K, log(g) = 5.5) with a temperature difference of 200K.

Though there is no definitive combination of colors to identify the age of an object irrespective of temperature, what I have done here is found a collection of prescriptions that are reliable indicators of age over small temperature ranges.

Color vs. Spectral Type Model Comparison

Here are color vs. spectral type plots for the 2MASS J, H and Ks bands.

The blue circles are for the objects with parallax measurements. The red squares are for the AMES-Dusty model spectra with spectral types gleaned from effective temperature according to Golimowski et al (2004).

While the AMES-Dusty models are known not to be a good fit for objects with effective temperatures lower than about 2200K (shown by the disagreement in L dwarf colors of objects and models), the M dwarfs fit fairly well for J-Ks versus Spectral Type.

However, the models are under-luminous in J-H and over-luminous in H-Ks for M dwarfs, indicating a possible problem with H-band modeling. The models shown are calculated with a surface gravity of 5.5, which means that the models produce a "peakier" H-band than the objects actually exhibit.

In a color-color plot of J-H versus H-Ks, the H-band throws off the model colors on both axes causing a diagonal shift (bluer in J-H, redder in H-Ks) of M and L dwarfs compared to the models: