In mathematics, the Laplace limit is the maximum value of the eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

 for which the series solution to Kepler's equation converges. It is approximately

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Kepler's equation

Kepler's equation

Kepler's equation is M = E -\epsilon \cdot \sin E ,where M is the mean anomaly, E is the eccentric anomaly, and \displaystyle \epsilon is the eccentricity....

 M = E − ε sin E relates the mean anomaly

Mean anomaly

In celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....

 M with the eccentric anomaly

Eccentric anomaly

In celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure...

 E for a body moving in an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

 with eccentricity ε. This equation cannot be solved for E in terms of elementary functions, but the Lagrange reversion theorem

Lagrange reversion theorem

In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions....

 yields the solution as a power series in ε:


Laplace

Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 realized that this series converges for small values of the eccentricity, but diverges when the eccentricity exceeds a certain value. The Laplace limit is this value. It is the radius of convergence

Radius of convergence

In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...

of the power series.