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Consider two sun-synchronous orbits with similar orbital elements, except for their times. For example 8h and 9h orbits. Their semi-major axes (a), inclinations (i), and eccentricities (e) are the similar and the orbits have no requirements of RAAN. (Though the rate of change of RAAN is same as they're both SSOs)

I am trying to find maneuvers that could bring a satellite from one of these orbits to the other. The way I look at it is that the satellite needs to catch up, with the other orbit, and shift its argument of perigee by ±15°. So that it reaches above a location, in ±1hr of the other.

The general approach I can think of is to change a, i or both so that the satellite goes into an intermediate orbit, and uses apsidal precession to achieve the ±1 hr orbit. I can see here how J2 affects the nodal and apsidal drifts. So I'm not sure how to define the intermediate orbit so that constraints on both ends are met.

Is there a prescribed way of solving such a problem, or maneuvers that are commonly used for this?

pathfinder_EOS
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    Are you trying to make the orbits intersect, or are you looking for a rendezvous? – GdD Apr 05 '22 at 16:05
  • There is no requirement to have them in the same or different planes. But given that the final orbit also needs to be an SSO with the same semi-major axis, that results in the same inclination. RAAN could differ as needed. – pathfinder_EOS Apr 05 '22 at 16:10
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    iirc, MRO (and other Mars assets) have done just this (shift ascending node LMST), there are some papers on the JPL TRS on their maneuvering – BrendanLuke15 Apr 05 '22 at 16:14
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    Plane changes (which is what you are asking about) are ridiculously expensive in terms of delta V. – David Hammen Apr 05 '22 at 17:06
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    @DavidHammen yes, plane changes are often very expensive. However, I was thinking that I could change the semimajor axis of the first orbit, changing its nodal drift rate, and when the required time is near, change the semimajor axis again to make it SSO. Does this make any sense or am I wrong in assuming that drift rate could be used to change SSO time? – pathfinder_EOS Apr 06 '22 at 10:03
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    changing the inclination will also change the drift rate – BrendanLuke15 Apr 06 '22 at 11:21
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    Thats right @BrendanLuke15, 'a' and 'i' both will change the drift rates as the equations in David Hamman 's answer here

    But as you can see in the same formulae, 'a' affects through a square term and 'i' affects through cos(i) term. Plus, for high inclination orbits such as SSOs, cos(i) is going to be a very small value. So overall the effects of changing 'a' might be much larger.

     – pathfinder_EOS Apr 06 '22 at 11:34
  • In addition to this, changing inclination is dV expensive and also requires attitude change as it requires out of plane burn. Therefore I thought that changing 'a' might be preferable.

    Does that sound reasonable?

    Also, am I fundamentally wrong somewhere in considering that drift rates could be useful in changing SSO times?

     – pathfinder_EOS Apr 06 '22 at 11:36
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    If you're up for some calculus you could find the partial derivatives $\frac{d\dot\Omega}{di}$ & $\frac{d\dot\Omega}{da}$ then multiply those by $\frac{di}{d\Delta V}$ & $\frac{da}{d\Delta V}$ to see which method is more dV efficient – BrendanLuke15 Apr 06 '22 at 12:28
  • I think you'll find the inclination is better because that derivative gets rid of the $\cos i$ – BrendanLuke15 Apr 06 '22 at 12:30
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    "...except for their times". Can you clarify if you meant orbital periods or local time when they pass the equator ? If they have same semi major axis and same eccentricities, wont they have same orbital periods (i.e. both 8h or both 9h) ? – AJN Apr 06 '22 at 13:18
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    @AJN I should've been more careful, I meant local times and not orbital periods. Orbital periods will be same. – pathfinder_EOS Apr 07 '22 at 08:46

1 Answers1

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Using the following equation (from David Hammen's answer) for RAAN rate of change:

$$\dot\Omega = - \frac{3}{2} J_2 \left(\frac R p\right)^2 n \cos i$$

Rearranged a bit:

$$Const. = -\frac{3}{2} J_2 R^2, n \propto \frac{1}{T}, T \propto a^{3/2}, p = a(1-e^2)$$

$$\dot\Omega \propto \frac{1}{a^{3/2}} \cdot \frac{1}{a^2(1-e^2)^2} \cdot \cos i$$ $$\dot\Omega \propto \frac{\cos i}{a^{7/2}(1-e^2)^2}$$

Finding the partial derivatives w.r.t inclination and semi-major axis will reveal that for a ~polar orbit (and probably most orbits) it is best to change inclination to affect nodal drift rate:

$$\frac{\partial \dot\Omega}{\partial i} \propto -\frac{1}{a^{7/2}} \cdot \sin i$$

$$\frac{\partial \dot\Omega}{\partial a} \propto \frac{1}{a^{9/2}} \cdot \cos i$$

BrendanLuke15
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    Calculus is scary, hopefully I've done this right – BrendanLuke15 Apr 06 '22 at 13:10
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    Wow thanks, this looks pretty solid! I'm going through the papers you previously pointed me to and they did infact change inclination, it makes sense. I will try this, and maybe some astro software too, to compare other effects. J2 affects drift of apsidal line as well and I think that will also change the time at which the spacecraft ascends across equator. I suspect using both drifts in the 'right' amounts will be cheapest in terms of dV. – pathfinder_EOS Apr 07 '22 at 07:57
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    Also could you please put a negative sign on the partial derivative wrt i? I can't edit since its just 1 character. – pathfinder_EOS Apr 07 '22 at 08:00
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    @pathfinder_EOS SSO's tend to be circular (e ~= 0) so argument of periapsis drift might not be a concern – BrendanLuke15 Apr 07 '22 at 11:12