Photometry

Measuring photometry in Lezargus is done using first principles and the following assumptions in this section. Most, if not all, of the filters we use are Vega-based filters, and so the treatment here assumes Vega-based filters. AB-magnitudes, where relevant, have different treatments which we also add briefly here.

See [[NOTE]]

Energy Counting and Photon Counting Convention

Photometric filter response curves found in the literature is often given as either energy integrating or photon counting conventions. More specifically, for photon counting detectors, like CCDs, there is an extra wavelength term to convert to the energy of a photon. As expected, they can be converted between each other:

\[\int F(\lambda) T_\gamma \mathrm{d}\lambda = \int F(\lambda) T_E \left(\frac{hc}{\lambda}\right)^{-1} \mathrm{d}\lambda T_\gamma \simeq T_E \lambda\]

More details can be found in Casagrande et. al. 2014. Overall, we follow their conventions here. We assume and keep the photometric filter response functions in their energy integrating form \(T_E\) and leave the wavelength component \(\lambda\) explicit. In the literature, some of the photometric response functions use the photon-counting formalism \(T_\gamma\); Lezargus converts them to the energy-integrating formalism to align with the procedures documented here. The photometry operations here are all done to wavelength-based spectra \(F(\lambda)\), with assumed units.

Photometric Standard Star

Photometry is a relative system, measurements of brightness are done relative to a photometric standard star. There are a number to choose from Landolt 2009, but, we pick Vega as our defining photometric standard star. We use the CALSPEC Vega spectrum (alpha_lyr_stis_011.fits); though should updated versions of said spectrum be available, we suggest updating to said versions. We cut the CALSPEC Vega spectrum to the wavelength limits that matter for Lezargus.

The relevant photometry magnitudes for Vega is as follows:

Johnson U 0.03 +/- ??? 2002yCat.2237….0D Johnson B 0.03 +/- ??? 2002yCat.2237….0D Johnson V 0.03 +/- ??? 2002yCat.2237….0D

Tycho2 BT Tycho2 VT 0.087 +/- ??? 2000A&A…355L..27H

GAIA G 0.029 +/- ??? 2018MNRAS.479L.102C GAIA BP 0.039 +/- ??? 2018MNRAS.479L.102C GAIA RP 0.023 +/- ??? 2018MNRAS.479L.102C

2MASS J -0.177 +/- 0.206 2003yCat.2246….0C 2MASS H -0.029 +/- 0.146 2003yCat.2246….0C 2MASS Ks 0.129 +/- 0.186 2003yCat.2246….0C

WISE 1 1.452 +/- 2012wise.rept….1C WISE 2 1.143 +/- 0.019 2012wise.rept….1C WISE 3 -0.067 +/- 0.008 2012wise.rept….1C WISE 4 -0.127 +/- 0.006 2012wise.rept….1C

Synthetic Magnitudes

The definition of magnitudes was set by convention from Pogson 1856 <https://doi.org/10.1093%2Fmnras%2F17.1.12>. Namely for a given filter profile \(T_E\), the magnitude \(m\) of a star \(F(\lambda)\) (relative to a photometric standard star \(F_0(\lambda)\) of magnitude \(m_0\)):

\[m - m_0 = -2.5 \log \left(\frac{\int F(\lambda) T_E\mathrm{d}\lambda}{\int F_0(\lambda) T_E\mathrm{d}\lambda}\right)\]

Fundamentally, as synthetic magnitudes are a weighted average of flux Casagrande et. al. 2014, it is appropriate to normalize over \(T_E\). Moreover, in the optical and infrared most detectors are photon-counting in nature (i.e. optical CCDs and infrared detectors Beletic et. al. 2008), we will adapt our form accordingly. (Conveniently, normalizing removes the \(hc\) constant.) The apparent magnitude relation can be rewritten as follows.

\[m - m_0 = -2.5 \log \frac{\int F(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda} + 2.5 \log \frac{\int F_0(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda}\]

Note, the factor of \(2.5\) here is technically \(10^\frac{2}{5}\), per Pogson 1856 <https://doi.org/10.1093%2Fmnras%2F17.1.12>, but most literature rounds said factor. The units must all be self-consistent else taking the logarithm with a unit is impossible Matta et. al. 2010 <https://doi.org/10.1021/ed1000476>. Otherwise, the full equation below must be used, combining the logarithms, so that the result is unitless.

\[m - m_0 = -2.5 \log \left[ \frac{\int F(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda} \left(\frac{\int F_0(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda} \right)^{-1} \right]\]

Zero Point Calculation

An alternative, but more useful and common, formulation of synthetic magnitudes start by defining the zero point \(Z\). (Other literature may have it noted as \(ZP\), but we use single letters here for clarity.) The zero point is defined solely by the filter profile \(T_E\) and the photometric standard star \(F_0(\lambda)\) and \(m_0\).

\[Z \triangleq m_0 + 2.5 \log \frac{\int F_0(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda} = m_0 + 2.5 \log \bar{F_0}\]

Note that \(\bar{F_0}\) is called the absolute calibration of the photometric system and is sometimes also called the zero point in some tables. Again, keeping in mind equivalent units. The zero point values published have units, and the unit system provided must be used for future calculations using that zero point. Most pertinent, is the determination of synthetic magnitudes.

Synthetic magnitudes using the zero point is calculated as follows, assuming standard units.

\[m = -2.5 \log \left( \frac{\int F(\lambda) T_E \lambda \mathrm{d}\lambda}{\int T_E \lambda \mathrm{d}\lambda} \right) + Z + \epsilon\]

Where \(\epsilon\) is a “fudge” factor sometimes added as a corrective term to replicate the definition of the photometric system more accurately. This may be done in cases where the photometric standard star given is not of sufficient quality.