The Laplace limit is the greatest orbital eccentricity for which Laplace's series solution of Kepler's equation converges.
For a complex number 
, the inequality
 |
(1)
|
describes the lens-shaped region illustrated above. In terms of real variables, the same region is described by
![1+lambda+sqrt(2(1+lambda-x^2+y^2))>=(x^2+y^2)exp[sqrt(2(1+lambda-x^2+y^2))],](/images/equations/LaplaceLimit/NumberedEquation2.svg) |
(2)
|
where
![lambda=sqrt([(1-x)^2+y^2][(1+x)^2+y^2]).](/images/equations/LaplaceLimit/NumberedEquation3.svg) |
(3)
|
This region can be parameterized in terms of a variable 
as
 |  |  |
(4)
|
 |  |  |
(5)
|
Written parametrically in terms of the Cartesian coordinates,
 |  |  |
(6)
|
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(7)
|
The parameter 
has maximum value 
(OEIS A085984;
Le Lionnais 1983, p. 36), the unique positive root
of
 |
(8)
|
or equivalently of
 |
(9)
|
Goursat (1959, p. 120) credits Stieltjes with the numerical value.
By symmetry, the area enclosed is therefore
 |  |  |
(10)
|
 |  |  |
(11)
|
Another exact reduction is obtained by defining 
and pairing two values 
and 
on the physical reflection sheet so that 
. With 
, this equation factors as
![S[(sinhS)/S(UV+1)-coshS]=0,](/images/equations/LaplaceLimit/NumberedEquation6.svg) |
(12)
|
where 
is defined to be 1 at 
.
The first component is 
,
or 
. Although explicit, it is not the
physical reflection sheet. At the cubic cusp 
, continuation of the physical area
switches instead to the nontrivial component
 |
(13)
|
On the nontrivial component, define 
and
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
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(17)
|
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(18)
|
For 
,
, while 
at 
. Together, these relations give 
, the ordinary transition between the 
and 
branches of the Lambert
W-function studied by Åhag et al. (2024).
Setting
 |  |  |
(19)
|
 |  |  |
(20)
|
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(21)
|
with 
for 
gives the exact area representation
 |
(22)
|
Equivalently, this is the explicit integral
 |
(23)
|
which agrees numerically with the parametrized formula above. Here 
is an associated
Laguerre polynomial, also called a generalized Laguerre polynomial. An exact
series representation for the area
is
 |  | ![(4pi)/(e^2)[1+sum_(k=1)^(infty)([L_(k-1)^((1))(4k+2)+L_k^((1))(4k+2)]^2)/((2k+1)e^(4k))]](/images/equations/LaplaceLimit/Inline70.svg) |
(24)
|
 |  |  |
(25)
|
(OEIS A140133). While the preceding representations are exact, they do not constitute a finite closed form free of both integration and an infinite sum.
This region is intimately related to the study of Bessel functions and Kapteyn series (Plummer 1960, p. 47; Watson 1966, p. 270).
The minimum value of 
corresponding to the maximum value
is 
(OEIS A033259;
Plummer 1960, p. 47; Watson 1966, p. 270), which is known as the Laplace
limit constant. It is the convergence threshold for
Laplace's series solution of Kepler's
equation and is the unique real number 
satisfying 
, where
 |
(26)
|
The same constant controls the existence of a catenoid spanning two equal coaxial circles of radius 
whose planes
are separated by a distance 
: such a catenoid exists only
for 
.
At equality it is the critical catenoid, and
the volume bounded by the catenoid
and its two boundary disks
is 
(Anderson 2026).
The continued fraction of 
is given by [0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, ...] (OEIS
A033260). The positions of the first occurrences
of 
in the continued
fraction of 
are 2, 10, 35, 13, 15, 32, 101, 9, ... (OEIS A033261).
The incrementally largest terms in the continued
fraction are 1, 27, 154, 1601, 2135, ... (OEIS A033262),
which occur at positions 2, 5, 11, 19, 1801, ... (OEIS A033263).
Branch Transition Function Arising from 
-Binomial Coefficients." Appl. Math. Comput. 462,
128347, 2024. 
Such That 
Using Library Functions?"
Mar. 12, 2022.