Question

In recent years, astronomers have found planets orbiting nearby stars that are quite different from planets in our solar system. Kepler-12b, has a diameter that is 1.7 times that of Jupiter, but a mass that is only 0.43 that of Jupiter. What is the value of g on this large, but low-density, world?

Answer 1

Answer:

The value of g in Kepler-12b is $3.85 m/s^{2}$

Explanation:

To determine the value of g at Kepler-12b it is necessary to combine the equation of the weight and the equation for the Universal law of gravity:

$W = m.g$  (1)

Where m is the mass and g is the value of the gravity

$F = G \frac{M.m}{R^{2}}$  (2)

Equation (1) and equation (2) will be equal, since the weight is a force acting on the object as a consequence of gravity:

$m.g = G \frac{M.m}{R^{2}}$  (3)

Then g will be isolated from equation 3:

$g = G \frac{M.m}{m.R^{2}}$

$g = \frac{G.M}{R^{2}}$  (4)

The radius of Jupiter has a value of 69911000 meters, so its diameter can be determined by:

$d = 2R$ (5)

Where d and R are the diameter and radius of Jupiter

$d_{jupiter} = 2(69911000 m)$

$d_{jupiter} = 139822000 m$

Procedure for finding the radius of Kepler-12b:

For the case of Kepler-12b it has a diameter that is 1.7 times that of Jupiter

$d_{Kepler-12b} = 1.7(d_{jupiter})$

$d_{Kepler-12b} = 1.7(139822000 m)$

$d_{Kepler-12b} = 237697400 m$

By means of equation 5 the radius of Kepler-12b can be known:

$R_{Kepler-12b} = \frac{d}{2}$

$R_{Kepler-12b} = \frac{237697400 m}{2}$

$R_{Kepler-12b} = 118848700 m$

Procedure for finding the mass of Kepler-12b:

The mass of Jupiter has a value of $1.898x10^{27}$ kilograms

For the case of Kepler-12b it has a mass that is 0.43 times that of jupiter

$m_{Kepler-12b} = 0.43(m_{jupiter})$

$m_{Kepler-12b} = 0.43(1.898x10^{27} Kg)$

$m_{Kepler-12b} = 8.161x10^{26} Kg$

The value of g in Kepler-12b can be found by replacing its radius and mass in equation 4:

$g = \frac{(6.67x10^{-11} N.m^{2}/Kg^{2})(8.161x10^{26} Kg)}{(118848700 m)^{2}}$

$g = 3.85 m/s^{2}$

Hence, in Kepler-12b the gravity has a value of $3.85 m/s^{2}$