In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun (Msun = 1.99 × 10^(30) kg) at a distance of 1.50 × 10^(11) m, called 1 astronomical unit (1 au). Others have extreme orbits that are much different from anything in our solar system. The following problem relates to one of these planets that follows circular orbit around its star. Assume the orbital period of earth is 365 days.

HD 10180g orbits with a period of 600 days at a distance of 1.4au from its star. What is the RATIOof the star's mass to our sun's mass?

So i know that Msun=1.99*10^(30).
And that Mstar/Msun gives me my answer. However I am having trouble getting the Mstar. I believe the distance of the Mstar is 6x10^11m and i got this by multiplying 1.5*10^(11) times (.4). After that I am not really sure what to do to get the mass? Please help!

The ratio of the star's mass to the sun's mass can be calculated using the equation Mstar/Msun = (2πr)^2/G(P^2), where r is the distance of the planet from the star, P is the orbital period of the planet, and G is the gravitational constant.

In this case, r = 1.4 au, P = 600 days, and G = 6.67 x 10^-11 m^3 kg^-1 s^-2.

Plugging these values into the equation gives us Mstar/Msun = (2π(1.4 au))^2/6.67 x 10^-11 m^3 kg^-1 s^-2 (600 days)^2 = 0.945 Msun.

Therefore, the ratio of the star's mass to the sun's mass is 0.945.

To find the mass of the star (Mstar), we can use the laws of planetary motion and the known values of distance and period.

The formula relating the period (T) of an orbiting object to the distance (R) from the center of the orbit is:

T² = (4π² / GM) * R³

Where:
T = Period
G = Gravitational constant (6.67430 × 10^(-11) m³ kg^(-1) s^(-2))
M = Mass of the central body (in this case, the star)
R = Distance of the planet from the star

We can rearrange this formula to solve for M:

M = (4π² / G) * (R³ / T²)

Now we can substitute the known values into the equation:

R = 1.4 AU = 1.4 * 1.50 × 10^11 m
T = 600 days
G = 6.67430 × 10^(-11) m³ kg^(-1) s^(-2)

First, let's convert the distance from AU to meters:
R = 1.4 * 1.50 × 10^11 m = 2.1 × 10^11 m

Next, let's convert the period from days to seconds:
T = 600 days * (24 hours / day) * (3600 seconds / hour) = 5.184 × 10^7 seconds

Now, substitute these values into the formula to find Mstar:

Mstar = (4π² / G) * (R³ / T²)
Mstar = (4π² / 6.67430 × 10^(-11) m³ kg^(-1) s^(-2)) * ((2.1 × 10^11 m)³ / (5.184 × 10^7 s)²)

Now, calculate this value using a calculator. The result will be the mass of the star (Mstar).

Finally, to find the ratio of the star's mass to our Sun's mass, divide Mstar by Msun:

Mstar / Msun = (Mstar) / (1.99 × 10^(30) kg)

Calculate this value, and you will have the ratio of the star's mass to our Sun's mass.

To calculate the mass of the star (Mstar), you can use the formula for circular orbital motion:

G * Mstar = (4 * π^2 * (Distance^3)) / (Period^2)

In this formula, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), Distance is the distance of the planet from the star (in meters), and Period is the orbital period of the planet (in seconds).

Let's convert the given values from astronomical units (au) and days to meters and seconds, respectively:

Distance = 1.4 au * 1.50 × 10^11 m/au = 2.10 × 10^11 m
Period = 600 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute ≈ 51,840,000 seconds

Now, substitute these values into the formula and solve for Mstar:

G * Mstar = (4 * π^2 * (2.10 × 10^11)^3) / (51,840,000)^2

Multiply both sides of the equation by (51,840,000)^2:

G * Mstar * (51,840,000)^2 = 4 * π^2 * (2.10 × 10^11)^3

Divide both sides of the equation by G and simplify:

Mstar = (4 * π^2 * (2.10 × 10^11)^3) / (G * (51,840,000)^2)

Now, substitute the value of G and calculate Mstar:

Mstar = (4 * π^2 * (2.10 × 10^11)^3) / (6.67430 × 10^-11 * (51,840,000)^2)

Mstar ≈ 4.066 × 10^29 kg

Finally, calculate the ratio of the star's mass (Mstar) to our Sun's mass (Msun):

Mstar / Msun = 4.066 × 10^29 kg / 1.99 × 10^30 kg

Mstar / Msun ≈ 0.205

Therefore, the ratio of the star's mass to our Sun's mass is approximately 0.205.