# What's the formula for the brightness of a signal mirror viewed at distance X, in space?

What's the formula for the maximum brightness of a triangle-shaped, flat signal mirror G with area A in orbit radius O around star S with brightness B and diameter D viewed at distance X, in space, viewed at angle θ?

By maximum brightness I mean, the angle is whatever gives the max brightness.

I'm not looking for "apparent brightness" to the human eye, I'm looking for the actual optical brightness as could be measured with a sufficiently powerful detecting instrument.

Ideally the brightness could be given in watts (signal strength). Thanks

Your variables are not very useful, which might be why this question wasn't answered already despite being fairly straightforward.

We want the irradiance $$E_e$$ (in $$W/m^2$$) due to a perfect mirror at distance $$r'$$, angle $$\theta '$$ from the mirror.

The mirror has area $$A$$ and is at distance $$r$$, angle $$\theta$$ from the sun, such that its cross-sectional area with respect to the sun is $$Asin(\theta).$$

Assuming a flat mirror, $$r >> \sqrt{A}$$, and $$r' >> \sqrt{A}$$, we see no image at all unless $$\theta ' = \theta$$ (angle of receiver is the same angular distance from the normal line of the mirror as the sun, and on the opposite side thereof).

So we are looking for $$E_e = f(r', r, A, \theta)$$

Power of the sun at its surface is that of an ideal blackbody at the star's surface temperature T, given by the Stefan-Boltzmann Law.

$$P = A_{sun}\sigma T^4$$

where $$\sigma = 5.67 \times 10^{-8}\frac{W}{m^2 K^4}$$

Irradiance at a distance r is just power divided by the surface area of a sphere of the same radius:

$$E_e = \frac{P}{4 \pi r'^2}$$

The radiant flux (in Watts) through the mirror is the cross-sectional area of the mirror multiplied by the irradiance at its radius.

$$\Phi_e = \frac{PAsin(\theta)}{4 \pi r^2}$$

Assuming a perfect mirror the radiant flux is all reflected, so

$$P_{mirror} = \Phi_e$$

You can take it from here. Irradiance from the mirror at radius r' is..? For a receiver of cross-sectional area A' the radiant flux is..?