Calculate the mass of the star using the fact
that the period of the planet is 3.07 × 107
s
and its distance from the star is 1.26×1011 m.
Answer in units of kg
The orbital period is given by
T = 2π√(r^3/GM)
We want M, which is thus
M = r^3/G * (2π/T)^2
Now just plug in your numbers.
To calculate the mass of the star using the given information, we need to apply Newton's version of Kepler's Third Law, which states that the square of the orbital period (T) of a planet is proportional to the cube of its average distance from the star (r).
The equation can be expressed as:
T^2 = (4π^2/GM) * r^3
Where:
T = orbital period of the planet (in seconds)
G = gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = mass of the star (in kg)
r = distance from the star to the planet (in meters)
We need to rearrange the equation to solve for M:
M = (4π^2/G) * (r^3 / T^2)
Now, we can substitute the given values:
T = 3.07 × 10^7 s
r = 1.26 × 10^11 m
G = 6.67430 × 10^-11 m^3 kg^-1 s^-2
M = (4π^2/6.67430 × 10^-11) * ((1.26 × 10^11)^3 / (3.07 × 10^7)^2)
Let's calculate the mass of the star using these values.
To calculate the mass of the star using the given information, we can make use of Kepler's third law, which relates the period (T) of a planet's orbit and the distance (d) between the planet and the star to the mass (M) of the star.
The equation for Kepler's third law is as follows:
T^2 = (4π^2/G) * (d^3/M)
Where:
T is the period of the planet's orbit (in seconds),
d is the distance between the planet and the star (in meters),
G is the gravitational constant (approximately 6.67430 x 10^-11 N(m^2/kg^2)),
M is the mass of the star (in kilograms).
Rearranging the equation gives us:
M = (4π^2/G) * (d^3/T^2)
Plugging in the values from the question:
T = 3.07 x 10^7 seconds
d = 1.26 x 10^11 meters
G = 6.67430 x 10^-11 N(m^2/kg^2)
We can now calculate the mass of the star using a calculator or a programming language:
M = (4π^2/G) * (d^3/T^2)
M = ((4 * 3.14159^2)/(6.67430 x 10^-11)) * ((1.26 x 10^11)^3)/(3.07 x 10^7)^2
After performing the calculations, the mass of the star can be obtained.