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ASTM001/MAS423 SOLAR SYSTEM. Carl Murray. Lecture 8: The Disturbing Function. DON’T PANIC!. Introduction. We have shown how the two-body problem can be solved. We have also looked at particular solutions of the three-body problem (equilibrium points, motion in their vicinity, etc.).
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ASTM001/MAS423 SOLAR SYSTEM Carl Murray Lecture 8: The Disturbing Function
DON’T PANIC!
Introduction We have shown how the two-body problem can be solved. We have also looked at particular solutions of the three-body problem (equilibrium points, motion in their vicinity, etc.). But how do we deal with the more general problem of an object (in orbit about a central mass) being perturbed by an additional mass? The disturbing function is simply the gravitational potential that one body exerts on another expressed in terms of a series involving the orbital elements of both bodies. Knowledge of the disturbing function allows us to understand how planets perturb one another and is the basis of our study of secular and resonant motion.
The Disturbing Function (1) Consider the motion of two masses orbiting a central mass.
The Disturbing Function (2) The equations of motion of the three objects, with respect to the origin are: The equations of motion of the two masses, with respect to the central mass are:
The Disturbing Function (3) Hence: These can also be written as: where direct part indirect part
The Disturbing Function (4) Now change notation so that the orbiting masses are m and m’ with position vectors r and r’ with The equation of motion for the inner mass is: and its disturbing function is: Similarly for the outer mass:
Expansion in Legendre Polynomials (1) The inverse of the separation can be expanded using Legendre polynomials:
Expansion in Legendre Polynomials (2) But, from the diagram, We also have, Hence: Note that the l=0 term does not involve r. Similarly,
Expansion in Legendre Polynomials (3) The expansion has the form: D’Alembert relation where By knowing the explicit form of the function S and the permissible combinations of the angles in j we can identify those terms that make the dominant contributions to the equations of motion and those that can be neglected. This is why it is so important to know the detailed form of expansion of the disturbing function and why so much effort has been devoted to it over the years.
Expansion in Legendre Polynomials (4) To illustrate some of the properties let us consider the planar case. Here the angle y is just the difference in the true longitudes of each mass: We can make use of the known series for sine and cosine of the true anomaly. This gives:
Expansion in Legendre Polynomials (5) Transforming from mean anomalies to mean longitudes: Note that (i) the absolute value of the sum of the coefficients of the mean longitudes gives the lowest order term in the eccentricity that can occur and (ii) the absolute value of the coefficient of the longitude of pericentre is the lowest power of eccentricity that occurs.
Expansion in Legendre Polynomials (6) Consider the radially dependent part of the disturbing function and write: Now consider just the l=2 terms in the summation:
Expansion in Legendre Polynomials (7) It is clear that it is a considerable effort to deal with just one term! Kaula (1962) developed a (complicated!) series expansion for the disturbing function:
Historical Expansions (1) Over the years several people have attempted to carry out series expansions of the planetary disturbing function. Benjamin Peirce published an expansion in the first edition of the Astronomical Journal in 1849.
Historical Expansions (2) Urbain Le Verrier published a 7th order expansion in 1855, using a compact notation.
Historical Expansions (4) One of the modern expansions is by Brouwer & Clemence (1961).
Historical Expansions (6) All these expansions have one thing in common: They are all in terms of the mutual inclination of the two masses. Note that this involves the nodes as well as the inclinations. The expansions also contained errors, some trivial, some serious. In the 1990’s Murray & Harper used computer algebra to carry out an expansion complete to 8th order in the individual orbital elements. The expansion and the typesetting was completely automated and numerous checks were done. The published expansion ran to 436 pages!
Historical Expansions (7) Increasing the order of the expansion increases the number of possible arguments that can occur. Because of the relationship between the mean longitudes and the powers of eccentricities, an expansion that includes nth order arguments will includes nth order terms in the eccentricities (and inclinations).
A Second Order Expansion (1) Here we give a complete expansion of the direct and indirect parts to second order in the orbital elements (and arguments). We can write the two disturbing functions as: where:
A Second Order Expansion (2) Laplace coefficients are functions of the ratio of semi-major axes (<1) and are uniformly convergent series for all a/a’<1. They can be written in terms of hypergeometric series.
A Second Order Expansion (3) Zeroth order arguments First order arguments Second order arguments Note again the connection between the order of the argument and the lowest powers of eccentricity that occur.
A Second Order Expansion (4) Note the absence of general arguments involving j.
Higher Order Expansions It is now possible to generate the specific terms associated with any valid argument using computer algebra. This makes it relatively easy to generate an expansion to any order! For example, Ellis & Murray (2000) derived the following formula for the direct part:
Using the Disturbing Function (1) It is clear that the expansion of the disturbing functin produces an infinite series. However, in practice we are only interested in particular cosine arguments (e.g. the ones appropriate near resonance) and we effectively neglect all (or most) other terms. The basis for doing this is the averaging principle, where we assume that all the non-important terms will be of short period and therefore will average out to zero over the longer period motion. We will see later how valid this is. The algorithm for dealing with the disturbing function is:
Lagrange’s Equations (1) Lagrange’s planetary equations relate the disturbing function to the rates of change of the orbital elements. They require the introduction of a new angle, e, the mean longitude at epoch. As explained in the notes, whenever e in a derivative it is to be interpreted as l. The variation in e itself is usually a small effect.
Classification of Arguments (1) For a given problem, arguments in the disturbing function can be classified as secular, resonant or short-period. Secular terms: These are terms which are independent of the mean longitudes. These can only come from zero order arguments with j=0.
Classification of Arguments (2) Resonant terms: Consider a general argument of the form We have: Hence: Therefore, if the semi-major axes are such that: then long-period effects will result. Therefore, the associated terms in the disturbing function are resonant terms.
Classification of Arguments (3) Short-period terms: Any terms not classified as secular or resonant (or near-resonant) are of short period and are invariably ignored. Section 6.9 of the notes gives further explanations of the distinctions between the different arguments. We will come back to these in Lectures 9 and 10 when we study secular and resonant motion. However, some numerical examples show how good the approximations can be:
