Does a satellite's Polar orbit revolve once around at the same rate the planet revolves around the Sun?

Take for example satellites in a Polar orbit around Mars.

Satellites in Martian Polar orbit follow the terminator, the line between sunlight and shade, in order to get the best quality images. Similar to the concept that when looking at the Moon you get the best views looking along the terminator line at First or Last quarter rather than at Full Moon when the sunlight is directed right down onto the surface at a steep angle and the surface appears shadowless.

In order for a satellite in Polar orbit to follow the terminator line entirely through the Martian year of 687 days, considering the terminator line will rotate completely once around through a complete revolution of Mars around the Sun (687 days), the satellite's Polar orbit itself would have to rotate at the same rate the terminator line does in order for the satellite to maintain an exact path over the terminator over the course of one Martian year.

So does the satellite's Polar orbit itself rotate at the same rate the planet's terminator line does? How is this possible?

I would think the Polar orbit maintains the same orientation throughout the Martian year and the terminator line revolves once per Martian year below the satellite's orbit.

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

If the Earth or Mars were perfectly spherical it would happen as you say. But the equatorial buldge causes the orbital plane of low satellites to drift in a predictable way. The orbital altitude controls the amount of drift, so an altitude is chosen such that the orbit's drift matches the terminator's drift over the course of a year. Look up "walking orbit".

** edit: sun-synchronous orbit as Grant suggests produces better Google results than "walking orbit"

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how can a bulge at the equator affect an orbit?

It distorts the gravitational field. The mathematical details are beyond my level, but I would imagine that there is a small and variable sideways component in the gravitational vector, compared to the perfect centrally directed force from a uniform sphere. I would welcome any comments from those who are more up to speed on the math.

Thanking you for the explantion and link suggestion grant hutchison and
tony873004. I've been following the NOAA sats for years. It's nice to know.
I have noticed that the polar orbiting sats don't actually fly directly over the poles. I think from memory (correct me please if I've got my foot in my mouth again.) the Russians perfected an early form of Sun synchronous orbit, the Molniya orbit. It went way out, then came close to the Earth, and yes retrograde.

The Molniya orbit is a bit different. The Molniya (Lightning) series were high-orbiting communication satellites. Since much of the territory of the USSR was at latitudes so high that geostationary satellites appear very low or in fact below the horizon, the next-easiest approach was an orbit which was very elongated and this spent most of its time near apogee, very high in the sky from northern latitudes and slowly moving. They adopted (IIRC) 12-hour periods for them, which keeps them close enough to Earth to minimize fuel neede for station-keeping against lunar and solar perturbations. (Ahh- Wikipedia informs me that both US and USSR did use Molniya orbits for reconnaissance purposes when dwell time or Sun angle was more important than ground resolution). These orbits are typically in about 63-degree inclinations, so not quite retrograde.

For low-orbiting reconnaissance satellites, the Soviets (and I think Russia continues to use this) found a "magic" orbit at high inclination, where the perturbations from the Earth's oblateness help counteract the slight drag of the atmosphere and allow a much longer lifetime in such a low orbit than one would normally expect. I'm having no luck at the moment in finding specific satellites in this orbit, despite having watched some over the years. There are just too many satellites names Cosmos and way too many google references with likely but useless terms...

As Hornblower says, the equatorial bulge stops the Earth behaving like a point source of gravity.
The additional gravitational pull from the equatorial bulge particularly affects the satellite as it passes through the equatorial plane of the Earth. The recurrent orbital tweaking caused by the extra mass at the equator makes the plane of the satellite's orbit precess in the opposite direction to the satellite's direction of motion: it's called regression of the nodes.

Grant Hutchison

Above what latitude are geostationary satellites not visible? Here in Canada we have people in very northern latitudes that used the Anik satellites for CBC tv reception, granted with some noise interference due to pointing their dishes so close to the "hot" earth!

I'm getting something like 85 degrees (although I wouldn't mind getting an independent confirmation from someone here), even if you're not very tall. That's assuming flat terrain, though.

I get a northern limit of about 81 degrees at the same longitude as the satellite, and farther south if too far away from the nearest satellite's longitude. Since I do not know their longitudes I cannot be more specific.