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Astronomy/Celestial mechanics question

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Expert: - 3/3/2010


Question
Can you provide a way to show the perturbation of planets without getting overly technical? (This is for my personal reference as a TA). I would like to see something like an order of magnitudes approach, say of the perturbation of any one planet (e.g. Earth) by another (say Jupiter) and also doing this from Legendre polynomials. Thanks!

Answer

Perturbation Diagram
Hello,

I have pretty well summarized the approach to be used from the figure I’ve uploaded.  The elements to this “perturbation” diagram include the respective masses: m1(Sun), m2(Earth) and m3 (Jupiter) and we assign relative radius vectors that approximate to the actual distance in AU. Thus, r = 1.0 and r3 (for Jupiter) = 5.  The Greek symbol DELTA defines the separation between Earth and Jupiter and is the key parameter for estimating the magnitude of the perturbation.

The angle S separating the r and r3 vectors can be anything but for working purposes maybe use S = 120 degrees, which yields a value: cos(S) = cos (120) = -  ½.  This will be useful for computing the first three Legendre polynomials – which factor into the kind of generic perturbations I am looking at.

To refresh, the first three Legendre Polynomials are:

P_o = 1

P_1 =  cos (S)

P_2 =    ½ (3 cos^2(S) – 1)


As we are perturbing an inner planet by an outer one, we are expanding (1/ Delta) using the Legendre polynomials.

Computing the Legendre polynomials for an Angle S = 120 deg, one finds:

P_o = 1     P_1 = - ½   and  P_2  = -1/8

From these you ought to be able to work out the relative magnitude using the summation of the perturbing function in the uploaded image, for example.

Further hint: A more standard form to obtain the estimate is:

R(p) = k^2 (m3)/  r3  {1 – ½(r/r3)^2 + 3/2 (r/r3)^2 cos^2 (S) + ……..+}

Bear in mind that k^2 = n^2a^3/ (m2 + m3)

Where n is the mean motion per day, and a is the “corrected semi-major axis”,  or a = 1.000000230 AU.

For more details including higher order Legendre polynomials and how they factor into the perturbations for a limited 3-body problem, consult Chapter 11, of ‘Modern Astrodynamics: Fundamentals and Perturbation Methods’ by Victor R. Bond and Mark C. Allman.

Philip Stahl

Expertise

I have forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics as applied to astronomical investigations.

Experience

Astronomy: more than forty years experience starting with construction of my own simple telescopes. Worked at university observatory in college, doing astrographic measurements. M.Phil. degree in Physics/Solar Physics and more than ten years as researcher.

Organizations
American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union

Publications
Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'.

Education/Credentials
B.A. Astronomy, M. Phil. Physics

Awards and Honors
American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research

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