How to calculate the light received by a planet during a binary star eclipse?

Question

I'm building up a spreadsheet on worldbuilding (Stars, Planets, Moons etc.) and i'm struggling to find an equation, or a group of equations to help me figure out the light received by a planet orbiting a P-Type Binary during both of its stellar eclipses.

I already have the light coming from the stars outside of the eclipses (represented in Joules), but searching for an equation to figure that out while one star is blocking the light of the other has alluded me. The closest i have come to finding an equation is one from the transit method, however that was intended to find the radius of exoplanets, so i don't think its quite relevant to what i'm looking for.

The specific numbers for the information needed to work this out isn't important - it's a spreadsheet, so they'll be changing/represented by cells. The specific units of measurement needed in this equation will be necessary however.

Answer 1

I'm making this up as I go along, so bear with me.

The light received by a planet when one star eclipses another depends on how much of the background star is blocked by the foreground star, which depends on the relative sizes of each and the line-of-sight angle from the planet, and on the brightness of each star, of course, and on the distance from the planet at each point.

This is rather complicated as it is, so let's assume the planet's orbital inclination is zero with respect to the star pair, i. e. the planet orbits on the same plane as the two stars so there will be an eclipse every time and it will be seen "head on". If the (apparent) larger star eclipses the (apparent) smaller one, of course, the solution is trivial.

In any case we need to know how large each star looks from the planet at the time of each eclipse. From a large distance $r$, the angular diameter of a body with actual diameter $d$ is $\delta = {r / d}$ (in radians).

Suppose stars have absolute brightness values $b_1$ and $b_2$. Units don't matter; we can express them relative to our Sun's brightness. Since the actual energy received will vary with the reciprocal of the square of the distance, you will have to calculate that. Again, we can choose relative units and measure distance in terms of the Earth-Sun system, e.g. treat distance as measured in AU.

For brightness $b$ and distance $r$, the planet will receive an effective amount of energy of $b / r^2$ (relative to Earth) from each star.

When there's an eclipse, the planet receives the full amount of energy from the foreground star ($b_1$) at its minimum distance ($r_1$), plus the energy corresponding to the part of the background star that is not eclipsed, if any. So this would be the answer to your question, expressed in terms of the relative brightness of both stars with respect to the Sun:

$$ b_{total} = {b_1 \over r_1^2} + \Delta\delta {b_2 \over r_2^2} $$

where $\Delta\delta$ is the relative difference between the angular diameter of the background star and the foreground star:

$$ \Delta\delta = {{r_2 d_1} \over {r_1 d_2}} - 1 $$

(as I said in the beginning, this is assuming the smaller star is passing in front of the larger one; if the reverse is true, $\Delta\delta = 0$). I'm using this because the difference between the angular diameters of each star determines how much of the background star is visible from the planet, and thus roughly how much of the star is sending photons towards it. I'm aware this might be a terribly crude of approaching it, but I don't think it wouldn't work as a nice first approximation.

Answer 2

This does not necessary provide the equations sought by the querent. There is an eclipsing binary simulator. It may not be possible to factor this into the spreadsheet. Sometimes other tools may be needed to do the job in another way.

It is possible to set the masses and surface temperatures of the two stars in the eclipsing binary pair. The light curve of the eclipse is generated and if this is treated as an approximation for the change in insolation, then it should be possible to derive an estimate for the light received by a planet.

The eclipsing binary simulator can be used empirically to construct a simplified model for the impact of a stellar eclipse on the amount of light received by the planet in this system. For example, it may be that the light received during a stellar eclipse is effectively the light received from the eclipsing star (since the light from the eclipsed star is de facto absent). Often where theory ends, experimentation has to fill the breach.

ADDENDUM:

After posting the above answer this author found the following information which does have equations about changes in flux from the stars in an eclipsing binary.

The light curve of eclipsing binaries gives information not only on the radii of the two stars but also on the ratio of their effective temperatures. This follows directly from eq. 2.13, L = 4piR^2sigmaT^4; as when an area piR^2 is eclipsed from the system, the drop in flux will be different depending on whether the hotter star of the two is in front or behind the cooler one (see Figure 4.6). Assuming for simplicity a uniform flux across the stellar disk,

we have:

F0 = A*(pi*Rl^2*Fl'+ pi*Rs^2*Fs') (4.16)

where F0 is the radiative surface flux, F0 is the measured flux when there is no eclipse, and A is a proportionality constant to account for the fact that we register only a fraction of the flux emitted (due to distance, intervening absorption and limited efficiency of the instrumentation). The deeper, or primary, minimum in the light curve occurs when the hotter star is eclipsed by the cooler one. In the example shown in Figure 4.6, this is the smaller star. Then, during the primary minimum we have:

F1 = A*(pi*Rl^2 * Fl' ---------- (4.17)

while during the secondary minimum:

F2 = A*(piRl^2 - piRs^2)*Fl' + A8pi8Rs^2*Fs' --------- (4.18)

To circumvent uncertainties in the constant A, we concern ourselves with the ratio of the two fluxes:

(F0 - F1)/ (F0 -F2) = Fs'/Fl' = (Ts/Tl)^4 ---- 4.19)

What eq. 4.19 tells us is that the ratio of the measured fluxes during the primary and secondary eclipses gives a direct measure of the ratio of the effective temperatures of the two stars in the eclipsing binary system.

Unfortunately, it wasn't possible to incorporate the equations into this answer without them turning into a bit of a mess. (See above mess) Please go to the source lecture for more information and for the equations in their proper form.

Source: Visual binaries