Mercury-Earth Conjunctions

Tonatiuh Sanchez-Vizuet and Matthew Moye, 17th June 2012

(Chebfun example linalg/MercuryEarthConjunctions.m)
[Tags: #linearalgebra, #rootfinding, #determinant]

The posititions of Earth and Mercury are given, relative to the sun at one foci of their elliptical orbits at (0,0), by the parametric equations [1]:

x_M(t) = -11.9084+57.9117*cos(2*pi*t/87.97)

y_M(t) = 56.6741*sin(2*pi*t/87.97)

x_E(t) = -2.4987+149.6041*cos(2*pi*t/365.25)

y_E(t) = 149.5832*sin(2*pi*t/365.25)

Conjunctions occur when Mercury is in a straight line configuration with the Earth and the Sun. A solution for the times of conjuctions can then be determined by the zeros of taking the cross product of the planets' position vectors on a time interval.

M = @(t) [-11.9084 + 57.9117 * cos(2*pi*t/87.97),...
           56.6741 * sin(2*pi*t/87.97);
          -2.4987 + 149.6041 * cos(2*pi*t/365.25),...
           149.5832 * sin(2*pi*t/365.25)];

f = chebfun(@(t) det(M(t)),[0 600],'vectorize');

The roots of the determinant give the days at which a conjunction occurs.

zeros = roots(f);

To get a visual interpretation of the roots, one can plot the determinant and then plot zeros all at value 0. The following figure depicts the times of the first ten conjunctions.

figure
plot(f), hold on
plot(zeros(1:10),0,'.r','markersize',10)
xlabel('Time(days)')

References:

[1] Charles F. Van Loan, Introduction to Scientific Computing, Prentice-Hall, 1997, p. 274.