Kepler’s Third Law – Planetary Motion Calculator

Kepler’s Third Law – Planetary Motion Calculator:

Planetary Motion Calculator is a tool that is used to find the satellite's orbital period using the mass of the planet and the radius between the position of the satellite and the selected planet.

What is Kepler's Third Law?

Kepler's Third Law tells that in how much time a satellite cover its one round about the planet using the relation between the orbital period and the distance of a satellite from the selected planet. Moreover, it is stated in the written form as the square of the orbital period is inversely proportional to the mass of a planet and directly proportional to the cube of the average distance between the planets.

Limitation of the Kepler’s Third Law:

Kepler's Third Law gives an important framework to understand the planetary motion but it has a few limitations. Factors such as the presence of gravitational influences and larger mass distributions of the planet can affect the accuracy of the orbital period.

These factors produced more difficulties that require additional factors and values to get a precise result. Moreover, it is a most significant contribution which tells that the orbital period and distance of a planet.

Formula of Kepler’s Third Law:

The mathematical formula for Kepler’s Law is stated as.

T = √4π² r³ /GM

It’s arranged to find the value of the Radius.

r = ³√T² GM /4 μ²

To find the value of the Mass of the Planet it becomes.

M = 4π² r³ / GT²

Where,

• T= Orbital period of the Satellite
• r = radius of the Satellite
• M = Mass of the Satellite
• G = gravitational strength of the earth

Note: The value of the gravitational Strength of the earth is “G= 6.67 × 10^-11” and the value of “π = 3.14” is used.

How to Find the Orbital period By Kepler’s Third Law:

Here solved some examples to understand the working of a tool in detail.

Example 1:

Evaluate the orbital motion of the satellite if the mass of the planet is 20 kg and the radius of the satellite is 4 meters.

Solution:

Step 1:

Write the data from the above conditions carefully.

T= Orbital period of the Satellite = ?,             r = radius of the Satellite = 70 m

M = Mass of the Satellite = 15 kg,                  G = gravitational strength = 6.67 × 10^-11

Step 2:

Write the formula of the orbital motion in detail.

T = √4π² r³ /GM

T = 2π × √ (r³ / (G × M))

Step 3:

Put the data in the above formula and simplify.

r = 70, M = 15, G = 6.67 × 10⁻¹¹

T = √4π² r³ /GM

T = 2π × √ (r³ / (G × M))

T = 2(3.14) × √ ((70)³ / ( 6.67 × 10⁻¹¹× 15))

T = 6.28 × √ (343000 / ( 0.05 × 10⁻¹¹))

T = 6.28 × √ (3428.29 × 10¹¹))

T = 6.28 × √ (342.829 × 10¹²))

T = 6.28 × 18.52 × 10⁶

T = 116.255505 × 10⁶s

T = 116255505.44011 s

Example 2:

If the Orbital radius of the satellite is 7 m and the Mass of the planet is 10 kg then find the orbital time period of the satellite.

Solution:

Step 1:

Write the data from the above conditions carefully.

T= Orbital period of the Satellite = ?,             r = radius of the Satellite = 7 m

M = Mass of the Satellite = 10 kg,                  G = gravitational strength = 6.67 × 10^-11

Step 2:

Write the formula of the orbital period in detail.

T = √4π² r³ /GM

T = 2π × √ (r³ / (G × M))

Step 3:

Put the data in the above formula and simplify.

r = 7, M = 10, G = 6.67 × 10⁻¹¹

T = √4π² r³ /GM

T = 2π × √ (r³ / (G × M))

T = 2(3.14) × √ ((7)³ / ( 6.67 × 10⁻¹¹× 10))

T = 6.28 × √ (343 / ( 66.7 × 10⁻¹¹))

T = 6.28 × √ (5.1424× 10¹¹))

T = 6.28 × √ (0.51424 × 10¹²))

T = 6.28 × 0.7171 × 10⁶

T = 4.5025563 × 10⁶s

T = 4502556 s