A comet orbits the Sun with a period of 72.0 yr.
(a) Find the semimajor axis of the orbit of the comet in astronomical units (1 AU is equal to the semimajor axis of the Earth's orbit).
(b) If the comet is 0.60 AU from the Sun at perihelion, what is its maximum distance from the Sun and what is the eccentricity of its orbit?
To find the semimajor axis of the orbit of the comet, we can use Kepler's third law, which states that the square of the period of an orbit is proportional to the cube of the semimajor axis.
(a) Given that the period of the comet's orbit is 72.0 years, we can set up the following equation:
T^2 = k * a^3
where T is the orbital period in years, a is the semimajor axis in astronomical units (AU), and k is a constant.
We can rearrange the equation to solve for the semimajor axis:
a^3 = T^2 / k
Since 1 AU is equal to the semimajor axis of the Earth's orbit, we can substitute T = 1 year and a = 1 AU into the equation to find the value of the constant k:
1^3 = 1^2 / k
k = 1
Substituting the values into the equation for the comet's orbit:
a^3 = (72.0)^2 / 1
a^3 = 5184
a ≈ 18.00 AU
Therefore, the semimajor axis of the comet's orbit is approximately 18.00 AU.
(b) The eccentricity of an orbit is given by the formula:
e = (R - r)/(R + r)
where e is the eccentricity, R is the distance from the focus to the farthest point on the orbit (maximum distance from the Sun), and r is the distance from the focus to the nearest point on the orbit (perihelion distance).
Given that the comet is 0.60 AU from the Sun at perihelion, we have r = 0.60 AU.
To find the maximum distance from the Sun, we need to express it in terms of the semimajor axis. The maximum distance occurs at aphelion, where the distance from the focus to the farthest point is a(1 + e).
Substituting the values into the eccentricity equation:
e = (a(1 + e) - r)/(a(1 + e) + r)
0.60 = (18.00(1 + e) - 0.60)/(18.00(1 + e) + 0.60)
Solving this equation will give us the value of e, the eccentricity of the orbit, and then we can find the maximum distance.
I apologize for not being able to provide the exact value of e and the maximum distance, as it requires solving a non-linear equation. You can use numerical methods or graphical methods to approximate the values.