Calculating the Synodic Period Between Earth and Mars

[sameapple]

Homework Statement

Homework Equations

From Wikipedia:

E = the sidereal period of Earth (a sidereal year, not the same as a tropical year)
P = the sidereal period of the other planet
S = the synodic period of the other planet (as seen from Earth)

The Attempt at a Solution

This problem requires me to actually "think" / make out / derive such an equation to calculate the time it takes for another opposition planetary alignment to re-occur between Earth and Mars. The problem, as shown in the diagram, gives me pure facts and I simply have no idea of how to start. For example, I do not know what are the important variables I must incorporate to compose such equation and I don't know how to mathematically relate the circular motion of Mars and Earth.

I have found an article on wikipedia showing me the way to calculate this so called synodic period which I believe is what I am looking for.

But, wikipedia doesn't tell me how exactly this equation is derived and I really want to learn the derivation. So anyone, please guide me.

 

[jbsteele]

The equation is: S = (E*P)/(E-P)where E = the sidereal period of Earth (a sidereal year, not the same as a tropical year) P = the sidereal period of the other planet S = the synodic period of the other planet (as seen from Earth)

 

[thomate1]

Calculating the synodic period between Earth and Mars is a complex mathematical problem that requires knowledge of celestial mechanics and orbital dynamics. The equation provided on Wikipedia is a simplified version and does not explain the full derivation.

To understand the derivation of this equation, one must first have a strong understanding of Kepler's laws of planetary motion and the concept of orbital periods. The sidereal period of a planet is the time it takes for the planet to complete one full orbit around the sun relative to the stars. The synodic period, on the other hand, is the time it takes for the planet to return to the same position relative to Earth.

To derive the equation for the synodic period, we must consider the relative motion of Earth and Mars. As Earth orbits the sun, Mars is also orbiting the sun, but at a different rate due to its larger orbit. This means that the two planets will not align in the same position every year, but rather at intervals of time. This interval is known as the synodic period.

To calculate the synodic period, we must consider the ratio of Earth's sidereal period (E) and Mars' sidereal period (P). This ratio will give us the number of times that Earth will have completed one orbit around the sun while Mars completes a certain fraction of its orbit. This fraction is known as the synodic phase (S).

The equation for the synodic period can be derived using this ratio and Kepler's third law, which states that the square of a planet's sidereal period is proportional to the cube of its semi-major axis (the average distance from the sun). By rearranging this equation and substituting in the ratio of the sidereal periods, we can get the final equation for the synodic period:

S = (P/E)^(2/3) - 1

This equation allows us to calculate the synodic period between any two planets. However, it is important to note that this is a simplified equation and does not take into account the gravitational influence of other planets and the elliptical shape of orbits. For more accurate calculations, more complex equations and simulations are needed.

In conclusion, calculating the synodic period between Earth and Mars requires a strong understanding of orbital dynamics and Kepler's laws. The equation provided on Wikipedia is a simplified version and does not explain the full derivation. To fully understand the derivation, one must have a strong background in celestial mechanics.