Kepler's equation

Last updated
Kepler's equation solutions for five different eccentricities between 0 and 1 Kepler equation solutions.svg
Kepler's equation solutions for five different eccentricities between 0 and 1

In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force.

Contents

It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova , [1] [2] and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation. [3] [4] The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics.

Equation

Kepler's equation is

where M is the mean anomaly, E is the eccentric anomaly, and e is the eccentricity.

The 'eccentric anomaly' E is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates x = a(1 − e), y = 0, at time t = t0, then to find out the position of the body at any time, you first calculate the mean anomaly M from the time and the mean motion n by the formula M = n(tt0), then solve the Kepler equation above to get E, then get the coordinates from:

where a is the semi-major axis, b the semi-minor axis.

Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically. Numerical analysis and series expansions are generally required to evaluate E.

Alternate forms

There are several forms of Kepler's equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits (0 e < 1). The hyperbolic Kepler equation is used for hyperbolic trajectories (e > 1). The radial Kepler equation is used for linear (radial) trajectories (e = 1). Barker's equation is used for parabolic trajectories (e = 1).

When e = 0, the orbit is circular. Increasing e causes the circle to become elliptical. When e = 1, there are three possibilities:

A slight increase in e above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as e goes to infinity, the orbit becomes a straight line of infinite length.

Hyperbolic Kepler equation

The Hyperbolic Kepler equation is:

where H is the hyperbolic eccentric anomaly. This equation is derived by redefining M to be the square root of −1 times the right-hand side of the elliptical equation:

(in which E is now imaginary) and then replacing E by iH.

Radial Kepler equation

The Radial Kepler equation is:

where t is proportional to time and x is proportional to the distance from the centre of attraction along the ray. This equation is derived by multiplying Kepler's equation by 1/2 and setting e to 1:

and then making the substitution

Inverse problem

Calculating M for a given value of E is straightforward. However, solving for E when M is given can be considerably more challenging. There is no closed-form solution.

One can write an infinite series expression for the solution to Kepler's equation using Lagrange inversion, but the series does not converge for all combinations of e and M (see below).

Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries. [5] Kepler himself expressed doubt at the possibility of finding a general solution:

I am sufficiently satisfied that it [Kepler's equation] cannot be solved a priori, on account of the different nature of the arc and the sine. But if I am mistaken, and any one shall point out the way to me, he will be in my eyes the great Apollonius.

Johannes Kepler [6]

Bessel's series expansion is [7]

Inverse Kepler equation

The inverse Kepler equation is the solution of Kepler's equation for all real values of :

Evaluating this yields:

These series can be reproduced in Mathematica with the InverseSeries operation.

InverseSeries[Series[M-Sin[M],{M,0,10}]]
InverseSeries[Series[M-eSin[M],{M,0,10}]]

These functions are simple Maclaurin series. Such Taylor series representations of transcendental functions are considered to be definitions of those functions. Therefore, this solution is a formal definition of the inverse Kepler equation. However, E is not an entire function of M at a given non-zero e. The derivative

goes to zero at an infinite set of complex numbers when e<1. There are solutions at and at those values

(where inverse cosh is taken to be positive), and dE/dM goes to infinity at these points. This means that the radius of convergence of the Maclaurin series is and the series will not converge for values of M larger than this. The series can also be used for the hyperbolic case, in which case the radius of convergence is The series for when e = 1 converges when M < 2π.

While this solution is the simplest in a certain mathematical sense,[ which? ], other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.

The solution for e ≠ 1 was found by Karl Stumpff in 1968, [8] but its significance wasn't recognized. [9] [ clarification needed ]

One can also write a Maclaurin series in e. This series does not converge when e is larger than the Laplace limit (about 0.66), regardless of the value of M (unless M is a multiple of ), but it converges for all M if e is less than the Laplace limit. The coefficients in the series, other than the first (which is simply M), depend on M in a periodic way with period .

Inverse radial Kepler equation

The inverse radial Kepler equation (e = 1) can also be written as:

Evaluating this yields:

To obtain this result using Mathematica:

InverseSeries[Series[ArcSin[Sqrt[t]]-Sqrt[(1-t)t],{t,0,15}]]

Numerical approximation of inverse problem

For most applications, the inverse problem can be computed numerically by finding the root of the function:

This can be done iteratively via Newton's method:

Note that E and M are in units of radians in this computation. This iteration is repeated until desired accuracy is obtained (e.g. when f(E) < desired accuracy). For most elliptical orbits an initial value of E0 = M(t) is sufficient. For orbits with e > 0.8, an initial value of E0 = π should be used. If e is identically 1, then the derivative of f, which is in the denominator of Newton's method, can get close to zero, making derivative-based methods such as Newton-Raphson, secant, or regula falsi numerically unstable. In that case, the bisection method will provide guaranteed convergence, particularly since the solution can be bounded in a small initial interval. On modern computers, it is possible to achieve 4 or 5 digits of accuracy in 17 to 18 iterations. [10] A similar approach can be used for the hyperbolic form of Kepler's equation. [11] :66–67 In the case of a parabolic trajectory, Barker's equation is used.

Fixed-point iteration

A related method starts by noting that . Repeatedly substituting the expression on the right for the on the right yields a simple fixed-point iteration algorithm for evaluating . This method is identical to Kepler's 1621 solution. [4]

functionE(e,M,n)E=Mfork=1tonE=M+e*sinEnextkreturnE

The number of iterations, , depends on the value of . The hyperbolic form similarly has .

This method is related to the Newton's method solution above in that

To first order in the small quantities and ,

.

See also

Related Research Articles

Eulers formula Complex exponential in terms of sine and cosine

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

Trigonometric functions Functions of an angle

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Hyperbolic functions Collective name of 6 mathematical functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t).

In mathematics, de Moivre's formula states that for any real number x and integer n it holds that

Legendre polynomials

In physical science and mathematics, Legendre polynomials are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

Orbital mechanics Field of classical mechanics concerned with the motion of spacecraft

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Chebyshev filters are analog or digital filters having a steeper roll-off than Butterworth filters, and have passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters".

Inverse trigonometric functions Arcsin, arccos, arctan, etc

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Trigonometric substitution Technique of integral evaluation

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.

Projectile motion Motion of launched objects due to gravity

Projectile motion is a form of motion experienced by an object or particle that is projected near the Earth's surface and moves along a curved path under the action of gravity only. This curved path was shown by Galileo to be a parabola, but may also be a line in the special case when it is thrown directly upwards. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards the Earth’s center of mass. Because of the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics.

Hyperbolic trajectory

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

Tangent half-angle formula Relates the tangent of half of an angle to trigonometric functions of the entire angle

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following:

In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.

Breather

In physics, a breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards an even distribution of initially localized energy.

Sine and cosine Trigonometric functions of an angle

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 that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .

Elastic instability

Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.

Kepler orbit

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal even though it is not an independent quantity and it relates to as .

The trigonometric functions for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:

References

  1. Kepler, Johannes (1609). "LX. Methodus, ex hac Physica, hoc est genuina & verissima hypothesi, extruendi utramque partem æquationis, & distantias genuinas: quorum utrumque simul per vicariam fieri hactenus non potuit. argumentum falsæ hypotheseos". Astronomia Nova Aitiologētos, Seu Physica Coelestis, tradita commentariis De Motibus Stellæ Martis, Ex observationibus G. V. Tychonis Brahe (in Latin). pp. 299–300.
  2. Aaboe, Asger (2001). Episodes from the Early History of Astronomy. Springer. pp. 146–147. ISBN   978-0-387-95136-2.
  3. Kepler, Johannes (1621). "Libri V. Pars altera.". Epitome astronomiæ Copernicanæ usitatâ formâ Quæstionum & Responsionum conscripta, inq; VII. Libros digesta, quorum tres hi priores sunt de Doctrina Sphæricâ (in Latin). pp. 695–696.
  4. 1 2 Swerdlow, Noel M. (2000). "Kepler's Iterative Solution to Kepler's Equation". Journal for the History of Astronomy . 31 (4): 339–341. Bibcode:2000JHA....31..339S. doi:10.1177/002182860003100404. S2CID   116599258.
  5. It is often claimed that Kepler's equation "cannot be solved analytically"; see for example here. Whether this is true or not depends on whether one considers an infinite series (or one which does not always converge) to be an analytical solution. Other authors claim that it cannot be solved at all; see for example Madabushi V. K. Chari; Sheppard Joel Salon; Numerical Methods in Electromagnetism, Academic Press, San Diego, CA, USA, 2000, ISBN   0-12-615760-X, p. 659
  6. "Mihi ſufficit credere, ſolvi a priori non poſſe, propter arcus & ſinus ετερογενειαν. Erranti mihi, quicumque viam monſtraverit, is erit mihi magnus Apollonius." Hall, Asaph (May 1883). "Kepler's Problem". Annals of Mathematics. 10 (3): 65–66. doi:10.2307/2635832. JSTOR   2635832.
  7. Boyd, John P. (2007). "Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial equation of the sine". Applied Numerical Mathematics. 57 (1): 12–18. doi:10.1016/j.apnum.2005.11.010.
  8. Stumpff, Karl (1 June 1968). "On The application of Lie-series to the problems of celestial mechanics". NASA Technical Note D-4460.{{cite journal}}: Cite journal requires |journal= (help)
  9. Colwell, Peter (1993). Solving Kepler's Equation Over Three Centuries. Willmann–Bell. p. 43. ISBN   0-943396-40-9.
  10. Keister, Adrian. "The Numerical Analysis of Finding the Height of a Circular Segment". Wineman Technology. Wineman Technology, Inc. Retrieved 28 December 2019.
  11. Pfleger, Thomas; Montenbruck, Oliver (1998). Astronomy on the Personal Computer (Third ed.). Berlin, Heidelberg: Springer. ISBN   978-3-662-03349-4.