What is the meaning of 'not perturbative accelerations' in astrodynamics?

[Rasalhague]

What does "not perturbative accelerations" mean in the following quote?

Bate, Mueller & White: Fundamentals of Astrodynamics, p. 11: "To simplify this equation [for the acceleration of a satellite due to gravity, with respect to the earth] it is necessary to determine the magnitude of the perturbing effects compared to the force between Earth and satellite. Table 1.2-1 lists the relative accelerations (not perturbative accelerations) for a satellite in a 200 n[autical] mi[le] orbit about the earth."

The table gives acceleration in G's due to the sun, the moon and each of the planets in the solar system, together with a value for acceleration due to the Earth's oblateness. E.g. sun 6x10^-4, moon 3.3x10^-6. From the context, and the expression "it is necessary to determine the magnitude of the perturbing effects", I'd expect the values in the table to be just that, perturbative effects. What distinction is it making? I thought perturbation, in this context, meant gravitational effects in addition to those described by a given simplified model such as, in this case, supposing that the only gravitational influence on the satellite was that of the Earth and that the Earth was perfectly spherical.

 

[Filip Larsen]

I think they are referring to the numbers shown being acceleration ratios (relative numbers) and not the accelerations themselves. At least, I read the "not the perturbative accelerations" as "not accelerations" and not as "not perturbative".

 

[Rasalhague]

Thanks, Filip. That's a relief! Sounds like it's just a weird wording, rather than that I've completely misunderstood the concept. So "not perturbative accelerations" here means that they are "perturbative accelerations expressed in multiples of g" (gravitational acceleration at the Earth's surface), rather than perturbative accelerations expressed in any of the other units they use, say feet per second² or "canonical units"?

(Correction to what I wrote: I see from a later chapter that they include all other forces under perturbation, not just gravitational ones.)

 

[Canticle]

I'm not convinced that's right guys. The main solar and planetary accelerations are known and calculable, that is; 'non perturbative', but there are often other accelerations which are not consistent and are not easily calculable. The Flyby anomaly is one, probably related to that from the Earth's planetary 'bow shock' of dark matter particles, the standing wave of which on the ecliptic polar is know to ebb and flow, and perhaps other (interplanetary) shocks. It can be serious as the loss a couple of probes Mars? - was put down to it by some. Even comets and asteroids can occasionally add small acellerations. It seems a bit like a small quantum uncertainty!

 

[D H]

Perturbative means "small", Canticle. The relative accelerations listed in that table are all small; i.e., much less than 1. They most definitely can be treated as perturbative.

The table is correct as written. You are just misreading it, Rasalhague. Accelerations, whether perturbative or not, have units of length/time². Those relative accelerations are unitless; they are not accelerations per se. They are accelerations scaled by the dominant acceleration term, which is of course the acceleration due to Earth gravity with the assumption of a spherical Earth. (IIRC, that table also includes effects from non-spherical Earth gravity; my copy of the book is at work.)

One key point of that table, and the related tables and figures in the text, is to determine what effects need to be modeled. For a vehicle in low Earth orbit, atmospheric drag is the dominant perturbative effect. The uncertainty in drag is rather high, so it is rather silly to model effects that are orders of magnitude smaller than drag. There is, for example, no reason to model the effects of solar radiation pressure on a vehicle orbiting 300 km above the Earth.

 

[Canticle]

Sorry DH. I was using Oxford English, where it means disturbed/confused. In this context it is of course also normally very small. NASA has however often been surprised how large it can be, (and it also seems responsible for the flat galactic acceleration curve, unless you prefer MOND).

 

[D H]

Not even wrong. First off, the topic of this thread is vehicles in low Earth orbit, period. Galactic rotation is off topic. Second, things like the flyby anomaly, if it exists, are by all definition tiny, tiny accelerations.

 

[Rasalhague]

So, they mean "the relative [perturbative] accelerations [conceived of as a dimensionless value] (not [the] perturbative accelerations [themselves])? And the distinction they're making is between

(1) a number, x, "relative acceleration" (i.e. "relative perturbative acceleration"), which when multiplied by g (defined as 9.8... m s^-2) gives the "perturbative acceleration", xg

and

(2) the "perturbative acceleration xg itself, on the other?

Actually they use a capital G, but I'm guessing it's defined the same way as g, which has dimensions of acceleration according to Fishbane, Gasiorowicz & Thornton: Physics for Scientists and Engineers and Tipler & Mosca's textbook of the same title. And various online lists of physical constants:

http://www.alcyone.com/max/reference/physics/constants.html
http://www.ebyte.it/library/educards/constants/ConstantsOfPhysicsAndMath.html

And here with a subscript n:

http://physics.nist.gov/cgi-bin/cuu/Value?gn|search_for=adopted_in!
http://web.mit.edu/3.091/www/constants.html
http://en.wikipedia.org/wiki/Physical_constant

Or is their G dimensionless, unlike g, and their acceleration due to gravity at the Earth's surface G ft s^-2, so that the distiction is between, say a dimensionless relative perturbative acceleration 10^-3 G, due to the Earth's oblateness, and the actual (dimensional) perturbative acceleration 10^-3 G ft s^-2? In other words, the warning is their way of saying that this G is dimensionless. Come to think of it, that might be more likely, as I'd have thought the other would go without saying.

I see now that they define a perturbation simply as "a deviation from some normal or expected orbit", so I suppose what counts as a perturbation will depend a bit on the context. In this case, Canticle, the table appears in a discussion of the gravitational perturbative effects of the sun, moon, the other planets and the Earth's oblateness on a satellite in near-earth orbit. Example perturbations mentioned in a later chapter include non-gravitational effects atmospheric drag, radiation pressure and thrust. On p. 9, a list of "other" forces, besides gravity, operating in the n-body problem is: drag, thrust, solar pressure, perturb. etc. The book was published in 1971, before the launch of Pioneers 10 and 11.

 

[D H]

You are talking about this page in Bate, Mueller & White, Fundamentals of Astrodynamics: http://books.google.com/books?id=UtJK8cetqGkC&pg=PA11#v=onepage&q&f=false

Look at the very first entry in that table: Earth acceleration = 0.89. The accelerations are obviously scaled by g, 9.80665 m/s².

The history of modeling planetary orbits is chock full of the use of perturbation techniques. Perturbation techniques are used widely throughout mathematics and physics. Suppose the differential equations that govern some system include one term that clearly dominates over all others and that an analytic solution would exist if only those other effects weren't present. That is exactly the case here. The simple two body problem does have an analytic solution, and those other terms are small. The next largest terms after the 0.89g from assuming a spherical Earth are those due to the Earth's oblateness (10^-3) and the Sun (6×10^-4).

 

[Rasalhague]

Okay then, so their G is the same as g = 9.80665 m s^-2 after all, and by "relative acceleration not perturbative acceleration" they mean perturbative acceleration expressed in units of g, rather than perturbative acceleration expressed in some other units such as m s^-2 or ft s^-2.

Thanks for your help, D H.