A335828 Numerators of coefficients in a power series expansion of the distance between two bodies falling freely towards each other along a straight line under the influence of their mutual gravitational attraction.

1, 1, 11, 73, 887, 136883, 7680089, 26838347, 14893630313, 1908777537383, 2422889987331397, 233104477447558811, 2430782624763507659, 14420190617640617313953, 4515429325405165295004389, 812454316441781379614873497, 166481868581561511154267399013

Offset: 1

Author

Amiram Eldar, Jun 25 2020

Keywords

• nonn
• frac

Comments

Consider two point objects with masses m_1 and m_2 that are starting to fall towards each other from rest at time t = 0 and initial distance r_0. Foong (2008) gave the solution for the distance as a function of time, r(t) = r_0 * f(t/t_0), where t_0 = sqrt(r_0^3/(G*(m1+m2))), G is the gravitational constant (A070058), and f(x) = 1 - Sum_{n>=1} c(n) * x^(2*n) is a dimensionless function. c(n) are the rational coefficients whose numerators are given in this sequence. The denominators are given in A335829. The collision occurs when f(x) = 0, at x = Pi/(2*sqrt(2)) (A093954), which corresponds to the time t = (Pi/(2*sqrt(2))) * t_0.

A similar expansion was given by Ernst Meissel in his study of the three-body problem in 1882. In Meissel's expansion the coefficients are c(n)/2^n.

Examples

			The series begins with f(x) = 1 - (1/2)*x^2 - (1/12)*x^4 - (11/360)*x^6 - ...
		

References

• Sudhir Ranjan Jain, Mechanics, Waves and Thermodynamics: An Example-based Approach, Cambridge University Press, 2016. See page 97.
• Ernst Meissel, Über Reihen, denen man bei der numerischen Lösung des Problems der Dreikörperproblems begegnet, wenn die Anfangsgeschwindigkeiten Null sind, in: Jahresbericht über die Realschule in Kiel: Während des Schuljahres 1881/82, A. F. Jensen, Kiel, 1882, pp. 1-11.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..243
• S. K. Foong, From Moon-fall to motions under inverse square laws, European journal of physics, Vol. 29, No. 5 (2008), pp. 987-1003, alternative link.
• Jaak Peetre, Ernst Meissel and the Pythagorean problem - the Drei-Körper-Problem in the Nachlass Meissel, draft, 1997.

Crossrefs

Cf. A070058, A093954, A202623, A335829 (denominators).

Programs

Mathematica

c[1] = 1/2; c[n_] := c[n] = (2*Sum[(n - k)*(2*n - 2*k - 1)*c[n - k]*c[k], {k, 1, n - 1}] - Sum[(n - m)*(2*n - 2*m - 1)*c[n - m]*c[m - k]*c[k], {m, 2, n - 1}, {k, 1, m - 1}])/(n*(2*n - 1)); Numerator @ Array[c, 17]
(* or *)
Quiet[-Numerator @ CoefficientList[AsymptoticDSolveValue[{y[x]*y'[x]^2 == 2*(1-y[x]), y[0] == 1}, y[x], {x, 0, 25}], x][[3;;-1;;2]]] (* requires Mathematica 11.3+ *)

Formula

a(n) = numerator(c(n)), c(1) = 1/2, c(n) = (2 * Sum_{k=1..n-1} (n-k)*(2*n-2*k-1)*c(n-k)*c(k) - Sum_{m=2..n-1} (n-m)*(2*n-2*m-1)*c(n-m) * Sum_{k=1..m-1} c(m-k)*c(k))/(n*(2*n - 1)).

c(n) ~ c_0 * n^(-5/3) * (Pi/(2*sqrt(2)))^(-2*n), where c_0 = (3*Pi)^(2/3) / (18*Gamma(4/3)) = 0.277587...