Context: I've been having a discussion with my vectors and mechanics professor (course Mathematics BSc) about a problem on a recent coursework. The following is the model I came up with. I'm not an astronomer, so bear with me.
- $S$ the Sun;
- $E$ the Earth;
- $P$ some other planet in the solar system;
- $O$ the position of an observer somewhere on $E$;
- $0<l<90$ the latitude of $O$;
- $\Pi_E$ the orbital plane of $E$, the $xy$ plane;
- $\Pi_P$ the orbital plane of $P$;
- $\Pi_O$ the plane tangent to $E$ at $O$;
- $\alpha$ the acute angle between $\Pi_E$ and $\Pi_P$;
- $\beta=90-l$ the acute angle between $\Pi_E$ and $\Pi_O$.
- The local time at $O$ is midnight;
- The date is the Summer solstice;
- The visual sizes of $P$ and $S$ are both non-zero but otherwise negligible;
- The projections of $E$, $S$ and $P$ onto the $xy$ plane are roughly but not exactly collinear;
- The solar system is Euclidean;
- Light travels in infinite straight lines.
Given these assumptions, I think that $P$ is visible from $O$ if and only if $\alpha>\beta$. Am I right?