How Do You Calculate the Mass of a Star Based on Its Planet's Orbit?

In summary, the mass of the distant star can be calculated using the formula v=(GM/r)^(1/2) and substituting in 2πr/T (in seconds) for v. The answer in units of kg is 4.187530248*10^30. Alternatively, the equation T2 = 4π2r3/GM can be used to manipulate and solve for the mass, which also demonstrates the origin of Kepler's laws.
  • #1
od943
5
0

Homework Statement


A distant star has a single planet circling it in
a circular orbit of radius 3.33 × 1011 m. The
period of the planet’s motion about the star
is 836 days.
What is the mass of the star? The
value of the universal gravitational constant
is 6.67259 × 10−11 N · m2/kg2.
Answer in units of kg.


Homework Equations


v = 2 π r / T
ac = v2 / r
Fc = mac
Kepler's laws?

The Attempt at a Solution


uhhhh...have i have no idea where to start
 
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  • #2
wait i figured it out...you use v=(gm/r)^(1/2) and substitute in 2(pi)r/T (in seconds) for v...then solve for M
the answer i ended up with was 4.187530248*10^30
 
  • #3
Nice.
 
  • #4
od943 said:
wait i figured it out...you use v=(gm/r)^(1/2) and substitute in 2(pi)r/T (in seconds) for v...then solve for M
the answer i ended up with was 4.187530248*10^30

Well done! And your method is exactly right, Kepler's laws being unnecessary in this case, though just as useful if you know how.

You can, of course, combined the equations to get...

T2 = 4.π2.r3/GM

...and manipulated that as needed. You can see where Kepler's law comes from then, too.
 

Related to How Do You Calculate the Mass of a Star Based on Its Planet's Orbit?

1. What is the "Planet Orbit Mass Problem"?

The "Planet Orbit Mass Problem" refers to the challenge of accurately determining the mass of a planet based on its orbital characteristics, such as its distance from its star and the duration of its orbit.

2. Why is it important to accurately determine a planet's mass?

Accurately determining a planet's mass is crucial for understanding its composition and formation, as well as its potential habitability. It also provides valuable information for studying the dynamics of the planet's orbit and interactions with other objects in its system.

3. What factors contribute to the difficulty of solving the "Planet Orbit Mass Problem"?

One of the main challenges is that the mass of a planet cannot be directly measured, and must be inferred through indirect methods. Additionally, factors such as gravitational interactions with other bodies and uncertainties in observational data can also complicate the calculation of a planet's mass.

4. How do scientists currently approach solving the "Planet Orbit Mass Problem"?

Scientists use a variety of techniques, such as radial velocity measurements, astrometry, and transit photometry, to gather data on a planet's orbit and its effects on its star. This data is then analyzed and compared to models and simulations to estimate the planet's mass.

5. What are some potential future advancements in solving the "Planet Orbit Mass Problem"?

Advancements in technology and observational techniques, such as the use of space-based telescopes and high-precision instruments, may lead to more accurate measurements and a better understanding of the factors that affect a planet's mass. Additionally, continued research and advancements in theoretical models and simulations can also contribute to solving this problem.

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