Understanding Newton's Version of Kepler's law. Find the orbital period in each case. Part A: A planet with twice Earth's mass orbiting at a distance of 1 AU from a star with the same mass as the sun.
To find the orbital period of a planet using Newton's version of Kepler's law, we can use the equation:
T = 2π√(a³/GM),
where T is the orbital period, a is the semi-major axis of the orbit, G is the gravitational constant, and M is the mass of the central star.
In this case, we are given that the planet has twice Earth's mass and is orbiting at a distance of 1 AU from a star with the same mass as the sun.
Step 1: Convert the distance to meters.
1 AU is equal to approximately 1.496 x 10^11 meters.
Step 2: Substitute the values into the equation.
a = 1.496 x 10^11 meters
G = 6.67430 x 10^-11 m³⋅kg⁻¹⋅s⁻² (gravitational constant)
M = mass of the star = mass of the sun = 1.989 × 10^30 kg
T = 2π√((1.496 x 10^11)³ / (6.67430 x 10^-11)(1.989 × 10^30))
Step 3: Simplify the equation and calculate the value.
T = 2π√(3.36 x 10^33 / (1.334 x 10^20))
T = 2π√(2.52 x 10^13)
T = 2π(5.02 x 10^6)
T ≈ 31,540,932 seconds
Therefore, the orbital period of the planet is approximately 31,540,932 seconds.
To find the orbital period, we can use Newton's version of Kepler's third law, which states that the square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit.
Kepler's third law equation can be written as: T^2 = k * r^3,
where T is the orbital period, r is the distance between the planet and the star (semi-major axis), and k is a constant.
In this case, we have a planet with twice the mass of Earth orbiting at a distance of 1 AU from a star with the same mass as the sun. The distance between the planet and the star is the semi-major axis of the orbit.
The mass of the star does not affect the orbital period, as Newton's law of gravitation only depends on the masses of the objects involved and the distance between them, not the mass of the star.
We know that the semi-major axis (r) is 1 AU. Therefore, we can rewrite the equation:
T^2 = k * (1 AU)^3.
Now, let's find the value of k. To do that, we can use the orbital period of Earth around the sun.
We know that Earth's orbital period is approximately 365.25 days. Converting this to years (since the units need to be consistent), we get:
T_earth = 365.25 days/365 days/year = 1.00068 years.
Substituting this value into the equation, we can solve for k:
(1.00068 years)^2 = k * (1 AU)^3.
k = [(1.00068 years)^2] / [(1 AU)^3].
Now, we can substitute the values into the equation to find the orbital period of the planet with twice Earth's mass:
T^2 = k * (1 AU)^3.
T^2 = [(1.00068 years)^2] / [(1 AU)^3] * (1 AU)^3.
T^2 = [(1.00068 years)^2].
Taking the square root of both sides, we find:
T = 1.00068 years.
Therefore, the orbital period of the planet with twice Earth's mass, orbiting at a distance of 1 AU from a star with the same mass as the sun, is approximately 1.00068 years.