In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Laplace limit is the maximum value of 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 ...
for which a solution to
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 ...
, in terms of a power series in the eccentricity, converges. It is approximately
: 0.66274 34193 49181 58097 47420 97109 25290.
Kepler's equation ''M'' = ''E'' − ε sin ''E'' relates the
mean anomaly ''M'' with the
eccentric anomaly ''E'' for a body moving in an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
with eccentricity ε. This equation cannot be solved for ''E'' in terms of
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
s, but the
Lagrange reversion theorem gives the solution 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 ε:
:
or in general
:
Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of ''M'' other than a multiple of π if the eccentricity exceeds a certain value that does not depend on ''M''. The Laplace limit is this value. It is the
radius of convergence of the power series.
It is also the maximum of the function
It is the unique real solution of the
transcendental equation
:
A
closed-form expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
in terms of
r-Lambert special function and an infinite series representation were given by István Mező.
History
Laplace calculated the value 0.66195 in 1827. The Italian astronomer
Francesco Carlini found the limit 0.66 five years before Laplace.
Cauchy in 1829 gave the precise value 0.66274.
See also
*
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 ...
References
* .
* .
External links
*
*
Orbits
Mathematical constants
Series (mathematics)
{{physics-stub