# 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.