It's called a "sun-synchronous" orbit, and it doesn't quite follow the terminator.
The orbital plane of a satellite in low orbit will precess because of the gravity of the equatorial bulge of the planet it's orbiting. If you choose the inclination of the orbit to the planet's equator properly, you can get the orbit to precess at the same rate as the planet goes around the Sun.
For the Earth, the sun-synchronous inclination in LEO is around 98 degrees: a retrograde, near-polar orbit which will see permanently low sun angles if you set it up with an initial track near the terminator.

There's a similar family of orbits for Mars, but I don't know the details.

Grant Hutchison