If a space probe were sent into orbit around the sun that brought it as close as 0.5 AU to the sun and as far away as 5.5 AU, what would its orbital period be?
Can I use Kepler's 3rd law for this? I'm not sure how to take into account both AUs...
Yes, you can use Kepler's third law, but instead of the radius (which of course only applies to circular orbits), you must use the average of the closest and the farthest distance from this sun.
That distance, which astronomers call the "semimajor axis" of the ellipse, is 3.0 AU in this case.
Great, thanks. :)
Yes, you can use Kepler's 3rd law to determine the orbital period of the space probe in this scenario. Kepler's 3rd law states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit.
In this case, the semi-major axis of the orbit will vary depending on the closest and farthest distances from the Sun. Let's call the closest distance 0.5 AU and the farthest distance 5.5 AU.
To use Kepler's 3rd law, we need to find the average distance from the Sun, also known as the semi-major axis (a). The semi-major axis can be calculated by finding the average of the closest and farthest distances:
a = (0.5 AU + 5.5 AU) / 2
a = 6 AU / 2
a = 3 AU
Now that we have the semi-major axis, we can use Kepler's 3rd law to find the orbital period (T). The equation can be written as:
T^2 = k * a^3
where k is a constant.
To find the value of k, we can use the period and semi-major axis of a known planet. We can use Earth as an example.
Earth's orbital period (T_e) is approximately 1 year, which is about 365.25 days. Earth's average distance from the Sun (a_e) is approximately 1 AU.
Substituting these values into the equation, we get:
(T_e)^2 = k * (a_e)^3
(1 year)^2 = k * (1 AU)^3
1^2 = k * 1^3
1 = k
Therefore, k is approximately equal to 1.
Now, we can use the value of k and the average distance (a) to find the orbital period (T) of the space probe:
T^2 = 1 * (3 AU)^3
T^2 = 1 * 27 AU^3
T^2 = 27 AU^3
Taking the square root of both sides, we get:
T = sqrt(27) AU
T ≈ 5.196 AU
Therefore, the orbital period of the space probe in this scenario would be approximately 5.196 years.