I am going to find an expression for the times when the earth is on the straight line between the moon and the sun, and for the times the moon is on the straight line between the earth and the sun.
Given:
- The Earth follows a circular orbit around the sun, counter clockwise.
- The Moon follows a circular orbit around the earth, counter clockwise.
- The sun is in the center of the xy-plane.
At t=0, both earth and moon lies on the positive x-axis with the earth closest to the sun.
Earths distance to sun, $r_e$ 150 (million kilometers)
- Moons distance to Earth, $r_m$ 0.384 (million kilometers)
- The Earth use $T_e$ 365 days around the sun
- The Moon use $T_m$ 27.3 days around the earth
Done so far:
Found expression of the Earths position around the sun:
$r_e (t) = (r_e*cos((2pi*t)/T_e),r_e*sin((2pi*t)/T_e))$
Found expression of the moons position around Earth:
$r_m (t) = (r_m*cos((2pi*t)/T_m),r_m*sin((2pi*t)/T_m))$
And an expression of the moons position around the sun:
$r(t) = (r_e*cos((2pi*t)/T_e) + (r_m*cos((2pi*t)/T_m), r_e*sin((2pi*t)/T_e) + r_m*sin((2pi*t)/T_m))$
Should i use parallel vectors? In that case, how do I proceed?
Thanks in advance.