If it takes a planet 2.8 x 10 to orbit a star with a mass of 6.2x 10 kg, what is the average distance between the planet and the star

The answer can be found be using the formula of orbital velocity v=√{ (G*M) / R}

Here G = Gravitational constant (G=6.67*10^-11 m³ kg⁻¹ s⁻²)

M = Mass of star

R = The distance between planet and the star

As we also know, v=ωR,

Here, ω = angular velocity

R = radius

ωR=√{ (G*M) / R}, now, ω=2π/T, where T is the orbital period of the planet

(2π/T) * R=√{ (G*M) / R}, we "put to the power of 2" both sides to get rid of the square root and get:

{ (2π/T) * R}²=[√{ (G*M) / R}]²,

(4π²/T²) * R² = (G*M) / R, we multiply by R:

(4π²/T²) * R³=G*M, now we solve for R and get:

R³ = (G*M*T²) / 4π², we take the third root to get R:

R=∛{ (G*M*T²) / 4π²}, inserting the values of time and mass (T = 2.8 x 10 s and Mass = 6.2x 10 kg)

R=∛{ (G*M*T²) / 4π²},

R=R=∛[{G=6.67*10^-11 x 6.2x 10 x (2.8 x 10^2) / 4x3.14²},

R=9.36*10^11 m

So this will be the average distance between the star and the planet.

9.36 * 10^11 m

Explanation:

The formula of general velocity can be used here = v=√{ (G*M) / R}.

G = gravitational constant or 6.67*10^-11 m³ kg⁻¹ s⁻²

M = Mass of the star

R = radius

To find v we use the formula v=ωR.

Here ω is the angular velocity; ω = ω=2π/T

Here T is the orbital period of the planet

The formula is arranged likewise: (2π/T) * R=√{ (G*M) / R}

Step 2: { (2π/T) * R}²=[√{ (G*M) / R}]²

Step 3: (4π²/T²) * R² = (G*M) / R

Step 4: (4π²/T²) * R³=G*M

Step 5: R³ = (G*M*T²) / 4π²

Step 6: R=∛{ (G*M*T²) / 4π²}

Step 7: Add your numbers to the formula and we get the answer.