Word Problem: Involving planets

In summary, extrasolar planets can be detected by observing the wobble they produce on their host stars. The radius of the star's orbit around the system's CM can be calculated using the formula ra= (mb)(rb)/ma. Using this formula, the radius of the star's orbit around the system's CM for a Sun-like star with a Jupiter-like planet is approximately 8.58E8. With the ability to detect angular displacements of 1 milliarcsec, astronomers can detect the wobble of a star located at a distance of 37 light years. This technique can be applied to about 800 stars within a distance of 37 light years, assuming a uniform distribution of stars in our region of the Milky
  • #1
nick21324neo
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Description: An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass revolves around its star of mass . If no external force acts on this simple two-object system, then its CM is stationary. Assume and are in circular orbits with radii and about the system's CM.

A) Find radius of the star's orbit about the system's CM. (answered)

ra= (mb)(rb)/ma

B)Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, and the planet has an orbital radius of 7.8×1011 . Determine the radius of the star's orbit about the system's CM. (answered)

*used formula above: ra=8.58E8

C)When viewed from Earth, the distant system appears to wobble over a distance of . If astronomers are able to detect angular displacements of about 1 milliarcsec (1/3600 of a degree), from what distance d (in light-years = 9.46E15) can the star's wobble be detected ? (answered)

d = 37 lightyears; I found the answer by using the ArcLength formula:

Arclength=Radius of Arc*theta

D) The star nearest to our Sun is about 4 lightyears away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?

Attempt for part D): I attempt this problem by using basic geometry

I determined the circumference by using the the distance of 4 light year and divided it by 2ra

The answer is found: 1.4E8 stars which is wrong. What am I doing wrong? It just seems to be a lot of stars.

Thanks
 
Last edited:
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  • #2
An exact calculation will be more complicated but for a reasonable approximation we can assume that the star density is 1 star within the volume of up to 4 light years distance (4/3 pi (4 ly)3). We don't have to calculate that volume, just the r3 dependence is important: Up to 37 light years we have (37/4)3 = 791 times the volume. We expect about 800 stars with a distance of up to 37 light years.

Gaia is currently taking data, it will detect wobbles of the order of 0.1 milliarcseconds. It is expected to find thousands to tens of thousands of planets that way.
 

1. What is a word problem involving planets?

A word problem involving planets is a mathematical question that requires the use of concepts and equations related to planets, such as their distance from the sun, orbital periods, and gravitational forces. These types of problems often involve real-world scenarios, such as calculating the travel time between planets or determining the mass of a planet based on its gravitational pull.

2. How do I solve a word problem involving planets?

To solve a word problem involving planets, it is important to first identify the given information and what is being asked in the problem. Then, use relevant equations and formulas to calculate the solution. It may also be helpful to draw diagrams or create tables to organize the information and make the problem easier to understand.

3. What are some common equations used in word problems involving planets?

Some common equations used in word problems involving planets include Kepler's Third Law, which relates the orbital period and distance of a planet from the sun, and the formula for calculating gravitational force between two objects, which takes into account the masses and distance between the objects. Other equations may involve calculating speed, acceleration, or energy related to planetary motion.

4. Can word problems involving planets be solved using units other than the standard metric units?

Yes, word problems involving planets can be solved using units other than the standard metric units. However, it is important to ensure that all units used in the problem are consistent and can be converted to one another if needed. For example, if the distance between two planets is given in miles and the gravitational force is given in pounds, these units can be converted to the metric units of meters and kilograms, respectively, to solve the problem.

5. What are some real-life applications of solving word problems involving planets?

Solving word problems involving planets can have practical applications in fields such as astronomy, astrophysics, and space exploration. For example, calculating the distance and speed required for a spacecraft to travel from one planet to another can help in planning missions to explore our solar system. These types of problems can also help scientists better understand the behavior and relationships between planets and other celestial bodies.

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