How to convert apparent visual magnitude to SI units?

Question

How do I convert the apparent visual magnitude of astronomical objects to SI units (lumens)?

Answer 1

I found this from Wikipedia:

$$\mathrm{illuminance} = 10^{(-14.18 - M_\mathrm{V})/2.5}~\mathrm{lx}$$

https://en.wikipedia.org/wiki/Illuminance#Astronomy

Looks to be right. I checked the formula with the luminous flux of Sol given by Wikipedia as 3.75×10²⁸ lumens, with an (extraterrestrial) luminance at 1 AU of $$\frac{3.75 \times 10^{28}~\mathrm{lm}}{4\pi~\mathrm{au}^2} = 133000~\mathrm{lm/m^2}$$ The formula above gives instead a (terrestrial) illuminance of 105700 lm/m² for an apparent visual magnitude of -26.74 (which makes sense since there would be less illuminance on the surface beneath an atmosphere).

https://en.wikipedia.org/wiki/Sun

Answer 2

An object brightness is normally measured as flux densities (aka. flam in the unit of erg/s/cm^2/Angstrom, or fnu in the unit of erg/s/cm^2/Hz). Flux density is the measurement of energy/time (i.e., power) passing through a unit area (i.e., /cm^2). Since energy/time/cm^2 might not be constant given wavelengths, therefore measuring it given a small wavelength width (i.e., /Angstrom or /Hz) would be more accurate treatment.

Magnitude measures the same brightness as flam or fnu, but mathematically constructed to be easier to be interpreted. The connection between flam (or fnu) with magnitude is:

mag = -2.5 * log10(flam) + CONST

The CONST (i.e., constant) is actually arbitrary for making the representation of mag easily interpreted. For example, if an object has flam around ~ 1e-10 erg/s/cm^2/A, you might set CONST = +25 for calibrating the zero magnitude equivalently to 1e-10 erg/s/cm^2/A. This is why we also call the CONST and the "zero-point magnitude."

Typically, the measurement of brightness is tied to a certain system, e.g., Vega or AB, therefore the CONST is not an arbitrary, but a certain number to satisfy the condition.

To answer your question, this will be the steps to follow to go from apparent magnitude to flam: 1. Identify the system the apparent magnitudes are referring to. 2. Get the equivalent CONST (or, zero point). 3. Use the equation to get flam. 4. Transform cgs to SI unit if necessary. 5. Get lumens by following another transformation which involves integrating the solid angles and wavelengths weighted by a sensitivity function (e.g., how human eyes are sensitive to each wavelength).