It take 89.2 years for a comet to travel around its elliptical orbit in which its perihelion is 0.670 AU. (a) the semi major axis of the comet in AU. (b) an estimate of the comet's aphelion, both in astronomical units (AU).
To find the answers to these questions, we can use Kepler's laws of planetary motion and the formula relating the semi-major axis to perihelion and aphelion distances.
(a) The semi-major axis (a) of an ellipse can be calculated using the formula:
a = (r_perihelion + r_aphelion) / 2
where r_perihelion is the distance from the center of the ellipse to the perihelion (closest distance to the Sun), and r_aphelion is the distance from the center of the ellipse to the aphelion (farthest distance from the Sun).
Given that the perihelion distance is 0.670 AU, we can substitute this value into the formula:
a = (0.670 AU + r_aphelion) / 2
Now, we need to find the aphelion distance.
Kepler's second law states that a planet/comet sweeps out equal areas in equal times. In this case, since the comet is moving in an elliptical orbit, it travels faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion).
Thus, we can assume that the time taken for the comet to travel from perihelion to aphelion is half of its total orbital period, which is 89.2 years. Therefore, the time taken from perihelion to aphelion is 89.2 years / 2 = 44.6 years.
We can now solve for the aphelion distance.
Using Kepler's third law, we know that the square of a planet's/comet's orbital period is directly proportional to the cube of its semi-major axis:
T^2 = k * a^3
Given that T (orbital period) is 89.2 years and a (semi-major axis) is still unknown, we can write:
(89.2)^2 = k * a^3
Solving for k, we divide both sides of the equation by a^3:
k = (89.2)^2 / a^3
Now, let's substitute the value of T and k into the equation for aphelion:
44.6^2 = ((89.2)^2 / a^3) * a
Simplifying, we have:
44.6^2 = 89.2^2 / a
To find the value of a, we can rearrange the equation:
a = 89.2^2 / (44.6^2)
Calculating this, we get:
a ≈ 2.251 AU
So, the semi-major axis of the comet's elliptical orbit is approximately 2.251 AU.
(b) Now, to estimate the comet's aphelion distance, we can substitute the value of the semi-major axis (a) into the formula we derived earlier:
aphelion distance = 2a - perihelion distance
aphelion distance = 2(2.251 AU) - 0.670 AU
aphelion distance ≈ 3.832 AU
Therefore, an estimate of the comet's aphelion is approximately 3.832 astronomical units (AU).
Thus, the answers are:
(a) The semi-major axis of the comet's elliptical orbit is approximately 2.251 AU.
(b) An estimate of the comet's aphelion is approximately 3.832 AU.