Moments of the Distance from the Force Center in a TwoBody
Kepler Orbit
(Or "Just How Far Away is the Sun Anyway?")
Al Lehnen  Jeremy Kessenich 
Mathematics Department  Mathematics Department 
Madison Area Technical College  Mount Horeb High School 
3550 Anderson Street  53704 305 S. 8th Street 
Madison, WI 53704  Mt. Horeb, WI 53572 
(608) 2466567  (608) 4372400 ext. 2115 
alehnen@madisoncollege.edu  kessenichjeremy@mhasd.k12.wi.us 
http://my.execpc.com/~aplehnen/al.htm 
Table of Contents:
Abstract
1.Introduction
2.The
Average over Arc Length
3.The
Average over True Anomaly
4.
The Average over Eccentric Anomaly
5.
The Average over Time
6.
A Comparison of the Different Methods of Averaging
7.
A Comparison between the Moments Averaged over Time and Astronomical Data
8.
Conclusion
References
Abstract:
This work examines the mean distance from the force center ("sun") at a focus
to a planet in a Kepler elliptical orbit. Four different methods of averaging
are presented: the average over arc length, the average over the central
angle about the force center (the "true anomaly"), the average over the central
angle about the center of the ellipse (the "eccentric anomaly"), and the
average over time. For each of the averaging methods the analysis actually
develops explicit formulas for the mean of the distance raised to an arbitrary
real power. These formulas are expressed as functions of the eccentricity
of the orbit. Connections and comparisons between the different averages
are developed. In particular, for integer exponents greater than 2, the
time averaged moment of the distance is always given exactly by a polynomial
in the eccentricity squared. Closed form expressions are given for the averages
over true and eccentric anomalies for any integer exponents. Results for
arbitrary real exponents and the results for the average over arc length
are presented as explicit power series in the eccentricity squared. Finally,
a comparison is made between the exact time averaged moments of a Kepler
orbit and astronomical data obtained from the United States Naval Observatory
web site. The agreement between observation and a Kepler orbit model is analyzed
and it appears that the difference is a function of the exponent on the distance.
In 1609 Johannes Kepler published his New Astronomy in which he formulated the first two of his famous three laws of Planetary motion [10, p. 122]. This was followed by the third law which appeared in the Harmonice Mundi in 1618. According to the first law the planets follow elliptic orbits with the sun at a focus. Unlike a circle, on an elliptical orbit the distance from the sun to the planet is not a constant. It varies from its minimum value at perihelion to its maximum value at aphelion. A natural question to ask then is "What is the average distance from the sun to the planet?". For the case of the earth this value is often [4, p. 245, 9] stated as 149,597,890 km or 92,955,819 miles. Recently there seems to have been a renewed interest in the problem of the average distance in a Kepler orbit and the definition of the astronomical unit [6, 19, 28]. But what is meant by the "average"? In this work we present four different but related "averages" and generalize the analysis to compute these averages for the distance raised to any power.
One way to analyze the problem is to take the center of the ellipse as the origin and align the xaxis parallel to the major axis so that perihelion has the coordinates (a, 0). The semimajor axis is a and the sun will then have the coordinates , where is the eccentricity of the ellipse. This is illustrated in Figure 1. From familiar results of analytic geometry the semiminor axis b is given by and the ellipse can be represented parametrically by the following equations.
with . (1)
In the Kepler orbit the parametric central angle E is sometimes called [26, p. 89], the "eccentric anomaly". Designate r as the distance from the focus at to the point generated by the angle E. An application of the distance formula gives the surprisingly simple result that
. (2)
The ellipse can also be represented in polar coordinates as
with .
The angle is measured from the x axis about the focus at and is called the "true anomaly". The distance from the "sun" to the position of perihelion is r_{0} . The major axis is the sum of r at with r at , so that
.
It follows that and the distance from the sun to the planet can be written as
. (3)
Figure 1
2. The Average over Arc Length
The mean value of r averaged over "the points" on the ellipse can be calculated by a rather simple procedure. From the symmetry of the ellipse about the xaxis, the average of r over the first and second quadrants equals the average of r over the entire ellipse. Now consider points on the ellipse in the first quadrant generated by central angles with i ranging from 1 to m. The reflection about the y axis of these points are generated by the central angles . This is illustrated in Figure 2.
The average value of r over the
2m points above the xaxis is then given by
. The symmetry of the angles and the definition of an ellipse require that
, a result which also follows immediately from equation (2). Thus, the "geometric"
average distance from a focus to points on an ellipse is just the semimajor
axis a. For this reason some authors [26, p. 87] refer to a
as the mean distance of the orbiting body.
Figure 2
To generalize and extend this result to arbitrary powers of r, the element of arc length along the ellipse is required. From equation (1) this is calculated as
.
The "geometric" average of r^{n}, the n’th moment of r, will thus be defined as the average of r^{n} with respect to arc length.
(4)
It is sufficient to use an upper bound of in the integrals since the ellipse is symmetric about the xaxis. The denominator in equation (4) can be expressed in terms of a complete elliptic integral of the second kind [27, pp. 517518, 22]. An alternative formulation is to represent the integral as an infinite series in the eccentricity squared. For the binomial expansion of gives the following absolutely convergent infinite series.
(5)
The "double factorial", defined as , is used to make formulas more concise. Integration by parts gives the standard reduction formula [20, p. T3]
, which in turn implies the recursion that . By an induction argument one concludes that
Combining equations (5) and (6) gives the explicit formula,
Here, F is Gauss’s hypergeometric function [1, p. 556, 25, 27, p. 281]. On the domain the integrals of odd powers of cosine are zero by symmetry. Therefore, the numerator of equation (4) can be expanded as a series where only even powers of cosine appear.
An explicit power series in the eccentricity for the nth moment of r would facilitate both the computation and the interpretation of results. To obtain such an expression expand the hypergeometric function that appears in the following series.
, where
.
For the special case of n = 1, a more careful use of equation (8) gives the result that
3. The Average over True Anomaly ( )
A second way to calculate the mean of r^{n} over an ellipse would be to sample r^{n} uniformly over the angle . From equation (3) this average is computed by the following formula.
Again the integral is defined only on because of the symmetry of the ellipse about the xaxis. For any real number p let the integral be defined as
The problem of evaluating when n is a positive integer is solved by noting some fundamental symmetries. First the substitution demonstrates that . A second, less obvious symmetry is stated as a theorem.
Theorem 1: For any real number p, .
Proof: Consider for the transformation (motivated by the Weierstrass substitution for rational functions of sine and cosine [20, p. 570])
.
The differential is then given by . The standard trigonometry identities that and give after some manipulation the following rather simple differential relation.
Now to evaluate the expression needs to be expressed in terms of z.
Hence, . The limits of integration transform as follows:
All of the "pieces are assembled" below..
Thus, the transformation confirms the stated symmetry.
From Theorem 1 and equation (13) for any positive integer n, .(14)
From equations (12) through (14) the angle averaged moments of the distance from the focus for any positive integer n reduce to the following closed form expressions.
(16)
(17)
In addition for any real n, Theorem 1 and equation (12) imply that
. (18)
The hypergeometric function, although not required, can be used to derive these same results. Indeed, can itself be expressed as a hypergeometric function.
In this formulation, Theorem 1 and the
symmetry expressed in equation (18) are just applications of the Euler transformation
[1, p. 559, 8, p. 1069,
24].
The results for small integer values of n are shown in Table 1. These formulas are not new. Indeed, some of them can be found in the existing literature [16, p. 54, 17, p. 63, 23, 28]. However, it appears that the complete and rather simple characterization of for integer n is not well known.
Table 1: Mean Moments of the Distance on an Ellipse from the Focus Averaged over the Angle















For the special case of n = 1, the mean value of r is just the semiminor axis, .The mean of r averaged over being less than a is certainly reasonable. Measured about the focus, half of the angles are on one side of the latus rectum where r is less than . Thus "small" values of r are "oversampled" when compared to the average taken over arc length. The standard deviation of r averaged over true anomaly is given by the formula
.
To lowest order in the eccentricity this is the same answer as the average over arc length.
4. The Average over Eccentric Anomaly (E)
Probably the easiest way to calculate the mean of r^{n} is to sample over the angle E. From equation (2) and the symmetry of the ellipse about the xaxis this average is computed by the formula,
.
From equation (11) and the symmetry that , it follows that
.
Theorem 1 implies a fundamental symmetry between the mean of r^{ n} and the mean of r ^{n}^{1}.
(19)
When n is any positive integer this symmetry and the binomial theorem establish the following formulas.
The relations between the average over
eccentric anomaly and the average over true anomaly
are obtained from equations (12) and (19).
Table 2: Mean Moments of the Distance on an Ellipse from the Focus Averaged over the Angle E















The mean value of r averaged over eccentric anomaly is identical to the mean value of r averaged over arc length. The standard deviation of r averaged over eccentric anomaly is given exactly by
.
For most people, once they have thought about it, the "average" distance of the earth from the sun would probably be calculated by measuring the distance every day for one full year and then dividing the sum of these measurements by the number of days in a year. That is, the average would be "over time", not arc length or angle about the sun. The result of a time average depends on the dynamics of the problem. If the earth orbited the sun at constant linear speed, then the average over time and the average over arc length would be identical. Similarly, if the earth orbited the sun at constant angular speed, then the average over time and the average over would be the same. However, for planets in elliptical orbits neither of these speeds is a constant.
The dynamics this work assumes is that of two spherically symmetric mass distributions interacting with each other through the inverse square law force of gravitational attraction. As is shown in many standard texts this results in the distance between the centers of the two bodies following the orbit of a conic section [7, pp. 5880, 20, pp. 857865, 26, pp. 7689]. For planetary speeds at perihelion which are less than the escape velocity, , the orbits are ellipses (this is Kepler’s First Law) with an eccentricity given by
. (22)
Here is the speed of the orbiting planet at perihelion, G is the "universal gravitational constant", M is the mass of the sun (or the "attractor") and m is the mass of the planet. The presence of m may come as a surprise to some readers. It is due to the separation of the motion of the center of mass from the twobody problem [7, pp. 5859, 26, pp. 7677]. This replaces the mass of the planet, m, in the Newton’s Second Law (F = ma) with the "reduced mass" . Since the sun is more than 300,000 times more massive than the earth, for the earthsun system M + m is often replaced by just M .
Suppose along the orbit there is a quantity, W, that can be written as an explicit function of , i.e., , then the timeaverage of W over one full revolution is given by the integral.
Here T is the period or time required for one full revolution. It is given by the formula
. (23)
This is really just the statement of Kepler’s Third Law. Since the equation of the elliptical orbit is given by equation (3) it is convenient to transform the variable of integration from t to .
Using the eccentricity as given by equation (22) results in a further simplification.
(24)
Thus, for a Kepler elliptical orbit there is a fundamental symmetry between the moments of r averaged over time and the moments of r averaged over true anomaly. This symmetry is summarized by the following brief formula, which is valid for any real exponent n.
(25)
In turn, this relation and equation (21) connect the moments of r averaged over time and the moments of r averaged over eccentric anomaly.
The symmetry of the average over true anomaly stated by equation (18) and the connection between the average over time and the average over true anomaly stated in equation (25) combine to reveal a corresponding symmetry present in the average over time.
Pasternack demonstrated in 1937 that there is an analagous symmetry between the expectation values of r^{n }and r^{}^{(n+3) } for the solutions of the nonrelativistic Schrodinger equation for the hydrogen atom [12].
From equations (15) and (25) if n is an integer and , then
From equations (16) and (25) if n is an integer and , then
. (27)
The resulting formulas for some small integer values of n are shown in Table 3.
Table 3: Mean Moments of the Distance on an Ellipse from the Focus Averaged over Time ( t )















Some of these results can be found in the literature [11, p. 51, 13, p. 38, 15, p. 41, 16, p. 54, 17, p. 63, 23, 28]. It should be noted that in most of these references the results would be inferred not as a time average, but rather as an average over a variable called the "mean anomaly" (for a definition see: [11, p. 14, 13, pp. 2425, 15, p. 18, 16, p. 47, 18, p. 152, 26, p. 89]).
However, as Serafin [14] has noted, the expectation value of a quantity as a function of time is identical to the expectation value of that same quantity as a function of mean anomaly. Surprisingly, the most concise and complete treatment of the time average of r^{n} we found is given not in an astronomy text, but in a treatise [3, pp. 143145] by Max Born on the “Old Quantum Theory”. If the complete and rather simple characterization of as stated in equations (26) through (28) are not new, then they are at least not very well known.
For the special case of n = 1, the mean value of r is . This is larger than a, the average over arc length. Since the planets move faster near the sun and slower when far away, they spend more time at long distances than at short distances. Thus, the time average must be somewhat larger than the spatial average. The desire to quantify this difference between the time average of r and a was the original motivation of this entire analysis.
The standard deviation of the time average of r is given by the formula
.
Thus, to first order in the eccentricity the four different ways of averaging all give the same result (the focal length over the square root of two) for the standard deviation of r. The corresponding equivalence for r^{n} will be demonstrated in section 6.
The result in Table 3 for is a specific example of a more general result for dynamical systems known as the "Virial Theorem"[7, pp. 6971]. The total energy is the sum of kinetic and potential energy terms and is a constant along a given orbit. Evaluating the total energy, E_{T}, at perihelion and using equation (22) gives the following result.
For a force that varies as the inverse
square, the Virial Theorem states that the timeaveraged potential energy
is twice the total energy. Hence,
or
.
6. A Comparison of the Different
Methods of Averaging
The original motivation for this work was the average distance (n = 1) from a planet to the sun. Table 4 summarizes the results already presented for this important case.
Table 4: Means and Standard Deviations of the Distance on an Ellipse from the Focus















Quite a few relations between the mean moments averaged over true anomaly, eccentric anomaly, and time have already been derived. However, it is very easy to get them confused. Table 5 provides a concise and explicit summary of the important exact symmetries of these three averaging methods.
Table 5: Exact Symmetries of the Moments of the Distance on an Ellipse from the Focus












An interesting application of the four averaging methods is the location of the mean position of a planet in a Kepler orbit. As elementary calculations based on equation (1) readily confirm, the mean position when averaged over either arc length or eccentric anomaly is the origin, i.e., the center of the ellipse. The average over true anomaly requires a little more effort.
The average value of y vanishes due to the symmetry of the ellipse about the xaxis, while the following trick establishes the average value of x.
Hence, for small eccentricities, the mean position of a planet averaged over true anomaly is approximately half way between the center of the ellipse and the focus (sun). Equation (24) provides the starting point to calculate the mean position averaged over time.
Again by symmetry, the average value of y is zero. Repeating the same trick as before computes the average of x.
For a Kepler orbit the timeaveraged position is exactly half way between the center of the ellipse and the “empty” focus opposite the force center. This result is given both by Plummer [13, p. 39] and Born [3, p. 145]. The distance between the mean position averaged over true anomaly and the mean position averaged over time is larger than the focal length of the ellipse but approaches the focal length for small eccentricities.
Further comparisons among the four different averages will be made at small values of the eccentricity so that terms of order and higher can be neglected. This is done for the following three reasons.
It makes the analysis particularly easy.
It is relevant. In our solar system all of the planets have orbits with eccentricities
less than 0.25 [9, 20, p. 864].
Differences between the three of the four methods of averaging are already
apparent in the terms of order
.
For any real value of n the moment equation
, (29)
follows from equation (9). An identical result is obtained from equation (20) for the average over eccentric anomaly. The binomial expansion applied to equations (17) and (28) gives
(30)
and
. (31)
From these equations a quick calculation of confirms that all four methods of averaging give a variance of r^{n}equal to and a standard deviation of given by . This implies that for small eccentricities and have the same standard deviation.
Comparing equations (29) and (31) gives an approximate symmetry between the time average and the average over arc length.
.
The average of r^{n} over "underestimates" the "true" or arc length average of r^{n} , while the average of r^{n} over time "overestimates" the answer. Therefore, one might expect the averages of these two estimates to be close to the arc length average. Using all three of the equations (29) through (31) demonstrates that this is indeed the case.
.
We therefore have the truly "tongue
twisting" conclusion that for small eccentricities the average of the average
of r^{n} averaged over true anomaly and the average of r^{n}
averaged over time is the average of r^{n} averaged over arc
length!
7. A Comparison between the Moments Averaged over Time and Astronomical Data
A final point of interest is to compare calculated for a Kepler orbit with observations of the earth’s actual motion about the sun. Using the MICA (MultiYear Interactive Computer Almanac) located on the United States Naval Observatory web site [21], the daily distance from the earth to the sun in astronomical units (AU) was computed for the calendar year June 28, 2002 through June 28, 2003. On each date calculation was performed for 00:00 UT (“Universal Time”). Figure 3 displays the results of these calculations.
Figure 3
Yearly means and population standard deviations for were computed using the 365 values of r from June 28, 2002 through June 27, 2003. To perform these calculations it is necessary to determine a value for a, the semimajor axis of the earth’s orbit. It is often claimed that a = 1 AU. However, there exists some confusion (certainly on our part) on this point. In his text Taff gives a value of a = 1.00000023 AU [18, p. 191]. The comprehensive Allen's Astrophysical Quantities states both that a = 1 AU [4, top of p. 12] and that a = 1.00000105726665 AU [4, bottom of p. 12]! These differences are of course quite slight, but for small eccentricities the discrepancy between and 1 will also be small. Finally, we decided that the best approach was to be selfconsistent within the MICA calculations. A numerical search of MICA determined that aphelion occurred on July 06, 2002 at 03:27 UT and perihelion occurred on January 04, 2003 at 04:53 UT. The computed values of r at these times give a = 1.000004292 AU and this is the value we used.
In order to compute for a Kepler orbit the eccentricity must be specified. The value for the eccentricity of the earth’s orbit found in many elementary texts is 0.0167 [20, p. 864]. Reference 4 quotes the extremely precise value of [4, p. 245], while the JPL’s web site states a value of 0.01671022. We eventually settled on for reasons that we discuss below. Of course, all of these values give essentially the same results for the average of r^{n} over time of a Kepler orbit. The computed values are displayed in Tables 6 and 7 for nonzero integer values of n between – 4 and 4.
Table 6: Comparison for the Earth of MICA Calculations with results of a Kepler Orbit

























Table 7: Comparison for the Earth of MICA Calculations with results of a Kepler Orbit

























Let
represent the yearly average of based on the MICA calculations and
represent the fractional deviation between this mean and
;
.
The values shown for both
and
are close to 1, so that the values of
are quite small. It is really the agreement between
and
that measures the how well the Kepler twobody orbit fits the MICA results.
From Tables 6 and 7, this agreement, while fair, is not entirely convincing.
In contrast, the agreement between the standard deviations of the Kepler
orbit and those of the MICA calculations seems much better, but then this
is to be expected. The standard deviation of the MICA results is given by
Figure 4
The linear or “nearlinear” deviation
of the Kepler orbit results from the MICA calculations appears rather startling.
We therefore attempted to understand it better. The average given by a definite
integral assumes continuous and uniform sampling over the domain of integration.
The average of the MICA results was based on 365 discrete scores. Thus, there
is a “continuity” correction. This could be reduced by say downloading MICA
calculations for every hour in a year rather than every day, but we were
eager to avoid this complication! We therefore decided to turn the problem
around. Rather than approximate the discrete average by an integral, we decided
to use the discrete MICA values of r to perform a numerical quadrature
of
and then compare this result directly against
. Since a full year needs to pass for the calculation of
, two additional MICA calculations are needed. One is obviously for June
28, 2003, the second is a bit more problematic. In the spirit of selfconsistency
it would be nice to define T directly from the MICA results. Unfortunately,
there are already enough deviations from a Kepler orbit to cause r
to be nonperiodic over a mean tropical year. The MICA calculations have r
at 22:40 on June 25, 2003 matching the value of r at 00:00 on June
28, 2002. We therefore used the published value that the number of days in
a year is given by
[7]. Therefore, the right end point of the domain of integration requires
r to be calculated at 06:09 on Jun 28, 2003. Thus, to perform the
numerical integration there are 367 discrete values of r computed by
MICA. For j less than 366 define Rj to be the value of
at
and let R_{366} be the value of
at t = T .
The trapezoid rule approximation to
is then given by the formula,
A Simpson’s rule approximation requires the messy job of fitting a parabola through the last three R values on the last two unequal intervals, and . The approximation to the integral is the rather complicated looking formula shown below.
The results of the two approximations are displayed in Table 8. From the convergence of the two methods neither increased sampling of r values nor more sophisticated quadrature methods would significantly improve the answers.
Table 8: Numerical Approximations to for the MICA Generated Values of r
Approximation  n = 1  n = 2  n = 3  n = 4 
Trapezoid Rule 
1.0001316895  1.0004028096  1.0008133848  1.0013634686 
Simpson’s Rule 
1.0001316896  1.0004028098  1.0008133852  1.0013634691 
Trapezoid Rule 
1.0000077460  1.0001549613  1.0004417092  1.0008680819 
Simpson’s Rule 
1.0000077459  1.0001549611  1.0004417089  1.0008680815 
The fractional deviations of from for integer values of n ranging from –5 to 6 are displayed in Figure 5. The discrepancies have been reduced by slightly more than a factor of 2 from the discrepancies associated with the discrete averages. Nevertheless, there remains a nearly linear (the correlation coefficient is 0.999823639 ) deviation of the MICA calculations from the Kepler twobody results. We are uncertain whether this is a “real” linear deviation present in the actual orbit of the earth, an artifact of the MICA calculations, a flawed assumption in our treatment of the results of the MICA calculations, or a combination of these factors. In any case, it can not be explained away as just a “continuity” correction.
Figure 5
Despite our fixation in trying to explain
small differences, we were actually impressed by how well the Kepler orbit
averaged over time models the MICA results (reality?). After all, the Kepler
twobody solution ignores the gravitational forces between the planets and
between planets and their satellites. Furthermore, the relevant mass distributions
are not perfectly spherical. Eventually even general relativity makes corrections
[26, p. 389].
This work has revisited one of the most
famous and influential problems in the history of science. The primary question,
"What is the average value of the distance between a planet and the sun raised
to a power?", is both easy to state and to understand. Surprisingly, within
the constraints of the Kepler twobody orbit, the problem can be satisfactorily
solved using only the methods typically taught in the introductory calculus
sequence. In particular, the systematic characterization of the time averaged
moments of r^{n} for integer n seems especially simple
and appealing. Similarly, the symmetries inherent in the four different methods
of averaging do not, initially at least, appear selfevident. The precise
statement and demonstration of these symmetries would by itself seem to be
a topic of interest. Given the importance and reputation of the Kepler problem
it would be extremely surprising if any of these results are really new.
Nevertheless, the detailed solutions presented here may help motivate introductory
calculus students and their instructors. There is always a need for
interesting but tractable examples. Just the existence of multiple ways of
calculating the average distance is an “eyeopener” for some students. Finally, the systematic examination of the
function, while a much more challenging exercise in celestial mechanics than
this study, might be a problem worth further investigation.
Acknowledgements
We would like to thank Professor Charles Goebel of the University of WisconsinMadison
Physics Department for many helpful comments. In particular, it was Professor
Goebel who pointed out that our Theorem 1 is a special case of one of the
Euler transformations of the hypergeometric function. He also suggested to
us the problem of the mean position. The reference to the work of Max Born
was kindly provided by Professor Turgay Uzer of the Physics Department at
Georgia Institute of Technology. Professor Lorenzo Curtis of the Department
of Physics and Astronomy at the University of Toledo provided references
to his own work as well of that of Pasternack and Williams.
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