Equation Of The Centre
   HOME

TheInfoList



OR:

In two-body, Keplerian
orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal ...
, the equation of the center is the angular difference between the actual position of a body in its
elliptical orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an orbital eccentricity, eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some or ...
and the position it would occupy if its motion were uniform, in a
circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...
of the same period. It is defined as the difference
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
, , minus
mean anomaly In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical ...
, , and is typically expressed a function of mean anomaly, , and
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values be ...
, .


Discussion

Since antiquity, the problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another. In calculating the position of the body around its orbit, it is often convenient to begin by assuming circular motion. This first approximation is then simply a constant angular rate multiplied by an amount of time. However, the actual solution, assuming
Newtonian physics Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods ...
, is an elliptical orbit (a Keplerian orbit). For these, it is easy to find the mean anomaly (and hence the time) for a given
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
(the angular position of the planet around the sun), by converting true anomaly \nu to "
eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit, the angle measured at the center of the ellipse between the orbit's periapsis and the current ...
": :E=\operatorname\left(\ \sqrt \sin \nu, \ e + \cos \nu \right) where
atan2 In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\ mu \arctan\left(\fr ...
(y, x) is the angle from the x-axis of the ray from (0, 0) to (x, y), having the same sign as y (note that the arguments are often reversed in spreadsheets), and then using
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...
to find the mean anomaly: :M=E-e\sin E If M is known and we wish to find E and f then Kepler's equation can be solved by
numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
s, but there are also
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
solutions involving
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
of M. In cases of small
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
, the position given by a truncated series solution may be quite accurate. Many orbits of interest, such as those of bodies in the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
or of artificial Earth
satellite A satellite or an artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body. They have a variety of uses, including communication relay, weather forecasting, navigation ( GPS), broadcasting, scient ...
s, have these nearly-
circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...
s. As eccentricity becomes greater, and orbits more elliptical, the accuracy of a given truncation of the series declines. If the series is taken as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in eccentricity then it fails to converge at high eccentricities. The series in its modern form can be truncated at any point, and even when limited to just the most important terms it can produce an easily calculated approximation of the true position when full accuracy is not important. Such approximations can be used, for instance, as starting values for iterative solutions of
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...
, or in calculating rise or set times, which due to atmospheric effects cannot be predicted with much precision. The
ancient Greeks Ancient Greece () was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity (), that comprised a loose collection of culturally and linguistically re ...
, in particular
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
, knew the equation of the center as '' prosthaphaeresis'', although their understanding of the geometry of the planets' motion was not the same. The word ''equation'' (
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
, ''aequatio, -onis'') in the present sense comes from
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
. It was specified and used by
Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of p ...
, as ''that variable quantity determined by calculation which must be added or subtracted from the mean motion to obtain the true motion.'' In astronomy, the term
equation of time The equation of time describes the discrepancy between two kinds of solar time. The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun ...
has a similar meaning. The equation of the center in modern form was developed as part of perturbation analysis, that is, the study of the effects of a third body on two-body motion.


Series expansion

In Keplerian motion, the coordinates of the body retrace the same values with each orbit, which is the definition of a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
. Such functions can be expressed as periodic series of any continuously increasing angular variable, and the variable of most interest is the
mean anomaly In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical ...
, . Because it increases uniformly with time, expressing any other variable as a series in mean anomaly is essentially the same as expressing it in terms of time. Although the
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
of , it is not an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
so a power series in will have a limited range of convergence. But as a periodic function, a
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
will converge everywhere. The coefficients of the series are built from
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
s depending on the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. Note that while these series can be presented in truncated form, they represent a sum of an ''infinite'' number of terms. The series for , the
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
can be expressed most conveniently in terms of , and
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
s of the first kind,Brouwer, Dirk; Clemence, Gerald M. (1961). p. 77. :\nu = M + 2\sum_^\infty \frac 1 s \left\\sin sM, where ::J_n(se) are the
Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
s and ::\beta=\frac\left(1-\sqrt\right). The result is in
radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
. The Bessel functions can be expanded in powers of by, :J_n(x) = \frac\left(\frac\right)^n\sum_^\infty(-1)^m\frac and by, :\beta^m = \left(\frac\right)^m\left +m\sum_^\infty\frac\left(\frac\right)^\right Substituting and reducing, the equation for becomes (truncated at order ), :\begin \nu \approx M &+ \left(2e - \frace^3 + \frace^5 + \frace^7\right) \sin M\\ &+ \left(\frace^2 - \frace^4 + \frace^6\right) \sin 2 M\\ &+ \left(\frace^3 - \frace^5 + \frace^7\right) \sin 3 M\\ &+ \left(\frace^4 - \frace^6\right) \sin 4 M\\ &+ \left(\frace^5 - \frace^7\right) \sin 5 M\\ &+ \frace^6\sin6M + \frace^7\sin7M + \cdots \end and by the definition, moving to the left-hand side, gives an approximation for the equation of the center. However, it is not a good approximation when is high (see graph). If the coefficients are calculated from the Bessel functions then the approximation is much better when going up to the same frequency (such as \sin7M). This formula is sometimes presented in terms of powers of with coefficients in functions of (here truncated at order ), :\begin \nu = M &+ 2e \sin M + \frace^2\sin 2M\\ &+ \frac(13\sin 3M - 3\sin M)\\ &+ \frac(103\sin 4M - 44\sin 2M)\\ &+ \frac(1097\sin 5M - 645\sin 3M + 50\sin M)\\ &+ \frac(1223\sin 6M - 902\sin 4M + 85\sin 2M)+ \cdots \end which is similar to the above form.Moulton, Forest Ray (1914). pp. 171–172. This presentation, when not truncated, contains the same infinite set of terms, but implies a different order of adding them up. Because of this, for small , the series converges rapidly but if exceeds the "
Laplace limit In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately : 0.66274 34193 49181 58097 47420 97109 25290. Ke ...
" of 0.6627... then it diverges for all values of (other than multiples of π), a fact discovered by Francesco Carlini and
Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
.


Examples

The equation of the center attains its maximum when the
eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit, the angle measured at the center of the ellipse between the orbit's periapsis and the current ...
is \pi/2, the true anomaly is \pi/2+\operatornamee, the mean anomaly is \pi/2-e, and the equation of the center is e+\operatornamee. Here are some examples:


See also

*
Celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
* Gravitational two-body problem * Kepler orbit *
Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force that varies in strength as the inverse square of the distance between them. The force may be either attra ...
*
Two-body problem In classical mechanics, the two-body problem is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are point particles that interact only with one another; th ...


References


Further reading

*Marth, A. (1890)
''On the computation of the equation of the centre in elliptical orbits of moderate eccentricities''
Monthly Notices of the Royal Astronomical Society, Vol. 50, p. 502. Gives the equation of the center to order ''e''10. *Morrison, J. (1883)
''On the computation of the eccentric anomaly, equation of the centre and radius vector of a planet, in terms of the mean anomaly and eccentricity''
Monthly Notices of the Royal Astronomical Society, Vol. 43, p. 345. Gives the equation of the center to order ''e''12. *Morrison, J. (1883)
''Errata''
Monthly Notices of the Royal Astronomical Society, Vol. 43, p. 494. {{DEFAULTSORT:Equation Of The Center Orbits