If a comet was at aphelion (furthest distance from the sun) in 1944 and will be at perihelion (closest point to the sun) in 2050, at a distance of 0.5 AU, what would it's aphelion distance be
To determine the aphelion distance of the comet, we need to calculate its average distance from the Sun, also known as the semi-major axis. Since the comet will be at perihelion in 2050 at a distance of 0.5 astronomical units (AU), we can use this information to find the semi-major axis by applying Kepler's third law of planetary motion.
Kepler's third law states that the square of the orbital period is proportional to the cube of the semi-major axis. It can be expressed as:
(T1^2 / T2^2) = (a1^3 / a2^3)
Where T1 and T2 are the orbital periods at two different distances, and a1 and a2 are the semi-major axes at those distances.
Now, let's assume that the comet completes one orbit every x years. We know the aphelion occurred in 1944, and the perihelion will occur in 2050, which is a time interval of 2050 - 1944 = 106 years.
Using the equation above, we can rearrange it to solve for a1, which represents the aphelion distance:
a1^3 = (T1^2 / T2^2) * a2^3
Since we don't know the specific value of T1, we can assume it is the same as the time it takes to go from aphelion to perihelion (1944 - 2050 = -106 years) because the comet's motion is symmetrical.
Thus, substituting the values into the equation, we have:
a1^3 = (-106^2 / 106^2) * (0.5^3)
Simplifying further:
a1^3 = 0.5^3
Taking the cube root of both sides:
a1 = cube root of (0.5^3)
a1 = cube root of (0.125)
a1 = 0.5 AU
Therefore, the aphelion distance of the comet is also 0.5 astronomical units (AU).