# Why should Molniya orbit period be a half-synodic, not a half-sidereal day?

In the Wikipedia article for Molniya orbit the introduction mentions that the period should be 0.5 sidereal day, so that its path across the sky every ~12 hours as observed from a given point on Earth remains constant.

But later in the properties section it refines that statement, saying that it should actually be 0.5 synodic day instead, and goes on to discuss the effect of the earth's oblateness upon the precession of the right ascension of the ascending node $$\Omega$$ and the argument of perigee $$\omega$$.

Since a geosynchronous orbit needs a period of 1.0 sidereal day to remain fixed in the sky, why is the Molniya referenced to a synodic day instead of a sidereal day?

Is there a simple way to understand this? Some kind of "Aha!" one-liner?

• Could it just be an error in the article? The Earth rotates, by definition, once every sidereal day relative to the vernal equinox. If the ground track of the satellite does not shift relative to the surface, its orbit should match that period as well. Sep 9 '16 at 23:49
• @JulianHzg There is something a bit confusing for sure. It seems the discussion is about adjusting the period to compensate for precession of some of the orbital elements, but I can't follow the logic. Also, I thought the difference between a sidereal and a synodic day is about 3m 56s, not the 35 seconds mentioned there.
– uhoh
Sep 10 '16 at 2:30

The wording is quite confusing. First off, the article frivolously uses the term 'orbital period' without ever qualifying which they mean: sidereal, synodic, draconic or anomalistic?

That changes the result, depending on how you understand the orbital period: Is it the time between the satellite reaching periapsis on two consecutive passes, or is it the time its true anomaly repeats?

First, a bit about the orbit. The Molinya orbit is to be used with constellations of three satellites; its purpose is to provide observation of two "interesting" regions of Earth in high latitudes; areas on two sides of Earth. This all from altitudes much lower than GEO (both lower distance for contact and observation, and lower launch costs), and thanks to the constellation configuration, 24/7 observation period is assured, at good angle to the surface of the "interesting" regions.

In particular, Soviets used it to observe Siberia and northern US.

The Molinya orbit has a very high eccentricity. The reason is to keep the satellites of the constellation for above the interesting area of Earth for the longest time (when they move slowly near the apoapsis) and let them dip into periapsis briefly when the interesting area moves out of focus (to be taken over by two next satellites of the constellation), only to return "into position" shortly later.

First, let's neglect orbital precession of the satellite. The optimal orbital period for this configuration is one sidereal day - the three orbits of the three satellites remaining in the same orientation relative to distant stars, and while the terminator line of Earth moves as Earth circles the Sun (and synodic day shifts relative to sidereal), the whole configuration remains constant relative to "distant stars", each satellite reaching apoapsis 1/6 sidereal day from the previous, and landing above the same location every sidereal day (and one on the opposite side of Earth, but same hemisphere, half a sidereal day later).

Therefore, in this configuration - taking the orbit without precession - it's accurate to say its period is half a sidereal day. That's also the number you get When you write down the orbital elements assuming Earth is a perfect sphere (as is commonly done) - so within this simplification, 0.5 "sidereal day" is accurate.

Now, Earth not being a perfect sphere, perturbs the orbit, causing its precession. The orbit was chosen such, that the perturbation per day is nearly exactly the angular shift between sidereal and synodic day, per day. As result, the orbit's argument of periapsis travels around Earth at one rotation per year, remaining in constant orientation relative to the Sun, and resulting in the orbit being synchronous with Earth surface at given time of synodic day.

It has little practical consequence, as both all three satellites retain their synchronicity and observation of the designated locations just the same, and the ground stations would have little trouble adapting to synodic period. Except the J2 term of the gravitational field of the Earth does inevitably cause some precession of any inclined, eccentric orbit - removing it would cost a lot of fuel, so instead the inherent error is transformed into a feature, the orbit chosen such, that it's synchronous to convenient, common sidereal time.

The angular precession per day is just a bit under 1 degree (360 degrees/365 days), and while the satellite completes the orbit the "sidereal" number of times - reaches the apoapsis 364 times per year - the apoapsis completes one full circle, each day both argument of periapsis and true anomaly of the satellite shifting by that 1 degree, adding one more orbital rotation to the number and resulting in the "synodic day period" result.

• Thanks! I'll take some time to read this - it is extremely helpful!! At the end of Properties it says "The rotational period of the Earth relative to the node (i.e. the synodic day) will therefore be only 86,129 seconds, 35 seconds less than the sidereal day which is 86,164 seconds." Isn't a synodic day (on average) 24 hours - 236 seconds longer than a sidereal day, not 35 seconds shorter? Or is this a different kind of synodic day?
– uhoh
Sep 13 '16 at 11:00
• @uhoh: Ah. Very silly liberal application of the term "sidereal day". In this context, "sidereal day" is applied in the satellite orbit's frame of reference - forget the Sun, treat the orbit as an invariable. "central body" relative to which Earth rotates. I honestly have no clue what purpose is this paragraph serving. [also - the gist of the answer is the second paragraph.]
– SF.
Sep 13 '16 at 11:27
• A sidereal day - 23h 56m 4s is indeed 86,164 seconds like it says. It's "The rotational period of the Earth relative to the node (i.e. the synodic day) will therefore be only 86,129 seconds, " part that bothers me. Oh - did you just mean to say "synodic day" and not "sidereal day* in your comment?
– uhoh
Sep 13 '16 at 11:55
• @uhoh: damn, now I'm totally confused. Yes, that's what I meant, and after you pointed it out, the paragraph completely ceased to make sense to me.
– SF.
Sep 13 '16 at 13:53
• ya, I know - I've been looking at the article now and again for weeks. OK - how about you change your answer to "...though may not be incorrect."and we'll keep at it. I can think about your answer in a half day from now.
– uhoh
Sep 13 '16 at 17:20