Find the period of revolution of an artificial planet if the semi-major axis of the planet’s elliptic orbit is greater than that of the Earth’s orbit by 24 x 10^6 km. It is given that the semi-major axis of earth orbit round the sum is 1.5 x 10^8 km.
The ratio of orbit semimajor axis lengths is
a2/a1 = 1.524*10^8/1.50*10^8 = 1.016
For the earth, the period is P1 = 365.25 days.
Use Kepler's third law to get the period for the artificial planet.
a^3/P^2 = constant
P2/P1 = (a2/a1)^3/2 = 1.0241
P2 = 374 days
To find the period of revolution of an artificial planet, you can use Kepler's Third Law of Planetary Motion. This law states that the square of the period of revolution (T) is proportional to the cube of the semi-major axis (a) of the planet's elliptic orbit.
Mathematically, Kepler's Third Law can be expressed as:
T^2 = k * a^3
Where T is the period of revolution, a is the semi-major axis of the planet's orbit, and k is a constant.
In this case, we are given that the semi-major axis of the Earth's orbit is 1.5 x 10^8 km. Let's assume the semi-major axis of the artificial planet's orbit is a + 24 x 10^6 km (where a is the semi-major axis of the Earth's orbit).
Substituting the values into Kepler's Third Law equation, we get:
T^2 = k * (a + 24 x 10^6)^3
To simplify the equation, we can divide both sides by k:
T^2 / k = (a + 24 x 10^6)^3
Now, we can rearrange the equation to solve for the period of revolution (T):
T = sqrt[(a + 24 x 10^6)^3 / k]
Since we don't have the specific value of k, we cannot calculate the exact period of revolution. However, if we assume that the value of k is the same for both the Earth and the artificial planet (which is a reasonable assumption), we can still proceed to find the ratio of the periods.
Let's denote T_earth as the period of revolution of the Earth and T_artificial as the period of revolution of the artificial planet.
Then, the ratio of the periods can be calculated as:
(T_artificial / T_earth)^2 = [(a + 24 x 10^6)^3 / a^3]
Now, let's substitute the given value of the semi-major axis of the Earth's orbit (a = 1.5 x 10^8 km).
(T_artificial / T_earth)^2 = [(1.5 x 10^8 + 24 x 10^6)^3 / (1.5 x 10^8)^3]
Simplifying this expression will give us the ratio of the periods (T_artificial / T_earth).
However, please note that without the value of the constant k, we cannot determine the exact period of the artificial planet's revolution.