Does anybody know how to solve this Earth-Venus-Sun problem?

Question

How can I find (in terms of the angle) the moment when in the system Sun-Earth-Venus, Venus can be seen the most bright if its brightness (flow received in Earth) is proportional to the size projected in its illuminated side? ¿What percentage of Venus's surface can be seen illuminated from Earth at that moment? We have to assume circular orbits for the Earth and Venus around the Sun and the distances between those planets and the Sun are known (1 UA and 0,723 AU)

Answer 1

Setting up the problem

Here is a little picture of our geometry (nothing to scale):

Here, $i$ is the phase angle, $v$ is the distance from the Sun to Venus, $e$ is the distance from the Sun to Earth and $d$ is the distance from Venus to Earth. Please pretend the line segments originate from the center of each body.

The brightness $b$ of Venus will be proportional to $p$ the percentage illuminated, given by Greg Miller in the notes as $p=\frac{1}{2}(1+\cos i)$. It will be inversely proportional to the square of the distance $d$ between Venus and Earth. We can express this as:

$$b \propto \frac{p}{d^2}$$

We just need to define $d$ as a function of $i$ and then we can find where $b$ is maximized.

Finding the distance as a function of Phase Angle

From the law of cosines, we have $$e^2=d^2+v^2-2dv\cos i$$ We can write this as a quadratic polynomial in $d$: $$d^2-(2v\cos i)d-(e^2-v^2)=0$$ We can solve for $d$ using the quadratic formula to get two solutions: $$d=\frac{2v\cos i \pm \sqrt{(2v\cos i)^2+4(e^2-v^2)}}{2}$$

We will get negative distance if we subtract the radical, so we can consider only the positive cases with a single solution of $d$ in terms of $i$ as:

$$d=\frac{2v\cos i + \sqrt{(2v\cos i)^2+4(e^2-v^2)}}{2}$$

Defining brightness as a function of phase angle

Plugging $d$ and $p$ back in to $b$ and simplifying, we get

$$b \propto \frac{p}{d^2} = \frac{2(1+\cos i)}{(2v\cos i + \sqrt{(2v\cos i)^2+4(e^2-v^2)})^2}$$

This now looks like a classic calculus problem, where we can take the derivative of $b$ with respect to $i$, set the result equal to zero and solve to find a value for $i$ which maximizes $b$. But why do calculus when you have a computer?

What phase angle yields maximum brightness?

Here is a graph of the (proportional) value of b through all phase angles in one degree increments:

Zero degrees is fully illuminated Venus at maximum distance from Earth. 180 degrees is fully shadowed Venus at minimum distance. The maximum brightness is at 119 degrees phase angle $i$. That is at about 26% illumination.

Note:

1. We make the additional assumption that the Earth and Venus orbits are coplanar.
2. I never make misnakes, but please feel free to check my work.