Keplers third law I understand.
T^2/a^3 of one planet = T^2/a^3 second planet.
But I can't do the algebra to solve for the second planet's a^3 (semimajor axis)
Would someone walk me through the steps please? Thanks.
a2^3= (T2/t1)^2 * a1^3
where a2 is the semimajor axis of the second planet.
Thanks
Certainly! I can help you with that. Let's go step by step to solve for the second planet's semimajor axis (a^3).
First, let's start with Kepler's third law equation:
T^2/a^3 of the first planet = T^2/a^3 of the second planet.
To solve for the second planet's semimajor axis (a^3), we need to isolate it on one side of the equation.
1. Start by cross-multiplying the equation:
T^2/a^3 of the first planet * a^3 of the second planet = T^2/a^3 of the second planet * a^3 of the first planet.
This simplifies to:
(T^2 * a^3 of the second planet) = (T^2 * a^3 of the first planet)
2. Next, divide both sides by T^2:
(T^2 * a^3 of the second planet) / T^2 = (T^2 * a^3 of the first planet) / T^2
This gives us:
a^3 of the second planet = (T^2 * a^3 of the first planet) / T^2
3. Now, cancel out T^2 on both sides of the equation:
a^3 of the second planet = a^3 of the first planet
This means that the semimajor axis (a^3) of the second planet is equal to the semimajor axis (a^3) of the first planet.
Therefore, according to Kepler's third law, both planets have the same value for their semimajor axis cubed.
Keep in mind that this equation assumes that the periods (T) of both planets are measured in the same units, such as years, and the semimajor axes (a) are measured in the same unit, such as astronomical units (AU).
I hope this explanation helps you understand the algebraic steps involved in solving for the second planet's semimajor axis using Kepler's third law.