Photometric Properties

In here, we describe bandpass and spectral photometric properties that can be calculated using pysynphot, along with their respective formulae. More information can also be found in Koornneef et al. (1986), Bandpass, and Observation.

The following table summarizes the available photometric properties, which are further elaborated in the sub-sections below:

Command	Description
bp.avgwave()	Bandpass Average Wavelength or reference wavelength
bp.rmswidth()	Bandpass RMS Band Width (Koornneef)
bp.photbw()	Bandpass RMS Band Width (SYNPHOT)
bp.rectwidth()	Bandpass Rectangular Width
bp.equivwidth()	Bandpass Equivalent Width
bp.efficiency()	Bandpass Dimensionless Efficiency
bp.unit_response()	Bandpass Unit Response
obs.effstim()	Effective Stimulus
obs.efflam()	Effective Wavelength
bp.pivot() obs.pivot()	Pivot Wavelength

These are not directly available but they are used to calculate other properties:

Bandpass Mean Log Wavelength
Bandpass Equivalent Monochromatic Flux

These are some common variables mentioned in the formulae in this section:

Variable	Description
\(F_{\lambda}\)	Source flux distribution
\(P_{\lambda}\)	Dimensionless bandpass throughput
area	Telescope collecting area
h	The Planck constant
c	The speed of light

Bandpass Average Wavelength

For a bandpass, avgwave() implements the equation for \(\lambda_{0}\) as defined in Koornneef et al. 1986 (page 836). It is equivalent to IRAF STSDAS SYNPHOT bandpar results for avglam, avgmw, or refwave; The throughput at this wavelength is tlambda.

\[\lambda_{0} = \frac{\int \; P_{\lambda} \; \lambda \; d\lambda }{\int \; P_{\lambda} \; d\lambda}\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.avgwave()
5373.2182275673349

Bandpass RMS Band Width (Koornneef)

For a bandpass, rmswidth() implements the bandpass RMS width as defined in Koornneef et al. 1986 (page 836), where \(\lambda_{0}\) is the Bandpass Average Wavelength.

\[\lambda_{rms} = \sqrt{\frac{\int \; P_{\lambda} \; (\lambda - \lambda_{0})^{2} \; d\lambda}{\int \; P_{\lambda} \: d\lambda}}\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.rmswidth()
361.9997795461671

Bandpass RMS Band Width (SYNPHOT)

For a bandpass, photbw() implements the equivalent for bandw from IRAF STSDAS SYNPHOT bandpar task, where \(\bar{\lambda}\) is Bandpass Mean Log Wavelength. This is not the same as Bandpass RMS Band Width (Koornneef).

\[bandw = \bar{\lambda} \; \sqrt{\frac{\int \; (P_{\lambda} / \lambda) \; \ln(\lambda \; / \; \bar{\lambda})^{2} \; d\lambda}{\int \; (P_{\lambda} / \lambda) \; d\lambda}}\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.photbw()
360.11107577076439

Bandpass Mean Log Wavelength

For a bandpass, this is the mean wavelength as defined in Schneider, Gunn, and Hoessel (1983). This rather unusual definition is such that the corresponding mean frequency is \(c / \bar{\lambda}\). This cannot be directly calculated by pysynphot, but is used for Bandpass RMS Band Width (SYNPHOT). It is equivalent to barlam in IRAF STSDAS SYNPHOT.

\[\bar{\lambda} = \exp\Bigg[\frac{\int \; (P_{\lambda} / \lambda) \; \ln(\lambda) \; d\lambda}{\int (P_{\lambda} / \lambda) \; d\lambda}\Bigg]\]

Bandpass Rectangular Width

For a bandpass, rectwidth() implements the rectangular width, where equvw is Bandpass Equivalent Width. It is equivalent to IRAF STSDAS SYNPHOT bandpar result for rectw.

\[rectw = \frac{equvw}{\mathrm{MAX}(P_{\lambda})}\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.rectwidth()
1124.6106504868569

Bandpass Equivalent Width

For a bandpass, equivwidth() implements the equivalent width. It is equivalent to integrate() and IRAF STSDAS SYNPHOT bandpar result for equvw.

\[equvw = \int P_{\lambda} d\lambda\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.equivwidth()
412.91237693252498

Bandpass Dimensionless Efficiency

For a bandpass, efficiency() implements the dimensionless efficiency. It is equivalent to IRAF STSDAS SYNPHOT bandpar result for qtlam.

\[qtlam = \int \frac{P_{\lambda}}{\lambda} d\lambda\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.efficiency()
0.077196835355538812

Bandpass Unit Response

For a bandpass, unit_response() implements the computation of the flux (in flam) of a star that produces a response of one count per second in that bandpass, where h and c are astronomical constants, and area is the telescope collecting area. It is equivalent to IRAF STSDAS SYNPHOT bandpar result for uresp.

\[uresp = \frac{hc}{area} \int P_{\lambda}\; \lambda\; d\lambda\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.unit_response()
1.9791581474812573e-19

Bandpass Equivalent Monochromatic Flux

For a bandpass, its equivalent monochromatic flux is as defined below, where \(\lambda_{0}\) is Bandpass Average Wavelength. It is equivalent to IRAF STSDAS SYNPHOT bandpar result for emflx.

\[emflx = \frac{uresp \times equvw}{P(\lambda_{0})}\]

This can be calculated indirectly in pysynphot, as given in the example below:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.unit_response() * bp.equivwidth() / bp(bp.avgwave())
2.3693354953649259e-16

Effective Stimulus

For an observation, effstim() calculates the predicted effective stimulus in given flux unit. countrate() is a special form of effective stimulus in the unit of counts/s given a pre-defined telescope collecting area. It is equivalent to IRAF STSDAS SYNPHOT calcphot result for effstim.

\[effstim = \frac{\int\; F_{\lambda}\; P_{\lambda}\; \lambda\; d\lambda}{\int\; P_{\lambda}\; \lambda\; d\lambda}\]

Example:

>>> obs = S.Observation(S.BlackBody(5000), S.ObsBandpass('acs,wfc1,f555w'))
>>> obs.effstim()  # photlam
0.00053965665649945897
>>> obs.effstim('flam')
1.9951166916464645e-15
>>> obs.effstim('counts')
10080.63299128226
>>> obs.countrate()
10080.633086603204

Effective Wavelength

For an observation, efflam() implements the effective wavelength, as defined in Koornneef et al. 1986 (page 836), where flux unit is converted to flam prior to calculations. It is equivalent to IRAF STSDAS SYNPHOT calcphot result for efflerg.

\[\lambda_{eff} = \frac{\int \; F_{\lambda} \; P_{\lambda} \; \lambda^2 \; d\lambda}{\int \; F_{\lambda} \; P_{\lambda} \; \lambda \; d\lambda}\]

Example:

>>> obs = S.Observation(S.BlackBody(5000), S.ObsBandpass('acs,wfc1,f555w'))
>>> obs.efflam()
5406.9723492971125

Pivot Wavelength

For an observation, pysynphot.observation.Observation.pivot() calculates the pivot wavelength. For a bandpass, it is pysynphot.spectrum.SpectralElement.pivot(). The formula below applies to an observation. For a bandpass, replace \(F_{\lambda}\) with \(P_{\lambda}\) in the formula. It is equivalent to IRAF STSDAS SYNPHOT result for pivwv and pivot.

\[\lambda_{pivot} = \sqrt{\frac{\int \: F_{\lambda} \; \lambda \; d\lambda}{\int(F_{\lambda} \; / \; \lambda) \; d\lambda}}\]

Example:

>>> bp = S.ObsBandpass('acs,wfc1,f555w')
>>> bp.pivot()
5361.007831073981
>>> obs = S.Observation(S.BlackBody(5000), bp)
>>> obs.pivot()
5394.930514954142