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I am currently trying to find the luminosity from the light curve. I have the data about the distance, counts/sec and the energy range in which it is measured for an object. What equation can be used to find the luminosity?

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    $\begingroup$ None. One energy range is not enough to fit a blackbody. You need measurements in at least two wavelength, or an unfiltered power / flux measurement, so that you have means to estimate the total flux from the source by assuming a blackbody radiator (or you need a different assumption like all power or a fixed percentage is emitted in your measured range) $\endgroup$ Oct 1, 2021 at 9:02
  • $\begingroup$ @planetmaker Not all astronomical objects are blackbodies. $\endgroup$ Oct 1, 2021 at 17:47
  • $\begingroup$ @planetmaker Also, if the goal is to determine the luminosity within a specified bandpass (rather than bolometric luminosity as you seem to be assuming), then you don't need observations in other bandpasses/wavelengths. $\endgroup$ Oct 1, 2021 at 18:01

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It's unclear what kind of luminosity you're trying to determine. If it's bolometric luminosity (from photons of all wavelengths), then planetmaker is right, and you need (calibrated) observations at multiple wavelengths.

Assuming, on the other hand, that you want to know the luminosity within "the energy range in which it is measured" (e.g., some filter bandpass, or even the narrow range of wavelengths spanning an emission line), then the relation between the observed flux $F$ and the luminosity $L$, given a distance $d$, is

$F = L / (4 \pi d^{2})$ .

The problem is that you don't have the flux, which should be in the appropriate physical units (ergs s$^{-1}$ cm$^{-2}$, W m$^{-2}$, etc.). You have an instrumental flux ("counts/sec"), which has to be transformed into physical flux units. To do that, you need a calibration (such as observing a source with known flux using the same telescope + instrument).

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