How can angular distance be used to measure the wobble of a star-planet system?

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In summary, if you measure the position of a star relative to a system's center of mass, you can determine if a planet is orbiting that system.
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eraemia
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Star-Planet System & Center of Mass

Homework Statement



Even with the best telescopes currently available, planets orbiting even the stars closest to the Earth are too dim compared to their parent stars to be imaged directly. However, one might indirectly detect a planet's presence by observing its gravitational effect on the star. Assuming that the star-planet system is very far from other stars, it will be essentially isolated, so its center of mass should move in a straight line. If the planet's mass is large enough, the system's center of mass will be displaced significantly from the star's center of mass. When the planet orbits the star, therefore, the planet and the star really both orbit the system's center of mass (like a pair of waltzing ballroom dancers), as shown in figure C4.6 (this figure is just an illustration, not necessary to solve this problem). We might therefore hope to detect a planet by observing how much a star's position "wobbles" around its general line of motion.

How difficult would this be? Assume that the star in question is a red dwarf that has a mass of about 0.30 times that of the sun (whose mass is 2.0 * 10^30 kg) and that the planet has 1.5 times the mass of Jupiter (whose mass is 1.9 * 10^27 kg) orbiting at a distance of about 1.5 * 10^12 m (about 10 times the distance from the Earth to the sun).

(a) About how far is the star's center of mass from the system's center of mass?

(b) If the star is 8.1 ly from us, what will be the star's maximum angular distance from the system's center of mass as seen by an earth-based telescope? Express your result in milliarcseconds, where 1 milliarcsecond = 1 mas = (1/3,600,000) degrees.

Homework Equations





The Attempt at a Solution



I don't even understand the problem... any hints appreciated!
 
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  • #2
What don't you understand? Can you draw a diagram showing the relevant parts of the problem (for yourself, that is -- no need to upload to here)? Do you understand the terms that the question uses?
 
  • #3
Okay, let me be more specific. I already solved (a). I knew the mass and position of the planet and the mass of the star. I set the center of mass of the star-planet system to 0, and then calculated the position of the star, which turned out to be: -7.1 * 10^9 m.

I'm confused by (b). What is angular distance? Is it asking the star's maximum angular distance from the STAR-PLANET SYSTEM or the earth-star-planet system? If it's the star-planet system, then wouldn't the answer be the same as (a)? So, what is angular distance, and which "system's center of mass" is the question referring to?

Thanks!
 
  • #4
eraemia said:
Okay, let me be more specific. I already solved (a). I knew the mass and position of the planet and the mass of the star. I set the center of mass of the star-planet system to 0, and then calculated the position of the star, which turned out to be: -7.1 * 10^9 m.

I'm confused by (b). What is angular distance? Is it asking the star's maximum angular distance from the STAR-PLANET SYSTEM or the earth-star-planet system? If it's the star-planet system, then wouldn't the answer be the same as (a)? So, what is angular distance, and which "system's center of mass" is the question referring to?

Thanks!

I am not 100% on this, but I believe it is asking you to express the distance (of the wobble?) from the center of mass of the star from the star-planet system.

the key is that it is the distance as seen from Earth. When you look up at night at the stars, it is almost meaningless to say "look at that star, the one that is one inch below the North star".

So instead we use angular distances i.e, "Look at that star that is 2 arcseconds south of the North star"

I don't know how accurate I was with the terms..but I think you get the concept.

Casey
 

What is a star-planet system?

A star-planet system is a group of celestial objects consisting of a star and one or more planets that orbit around it. The star provides light and heat to the planets, and the planets revolve around the star due to its gravitational pull.

How do stars and planets form in a star-planet system?

Stars and planets form from the same cloud of gas and dust called a nebula. The gravitational force within the nebula causes the gas and dust to clump together and form a protostar. As the protostar grows in size and temperature, it eventually becomes a star. At the same time, the material surrounding the protostar begins to coalesce into planets.

What types of planets can exist in a star-planet system?

There are four main types of planets that can exist in a star-planet system: terrestrial planets, gas giants, ice giants, and dwarf planets. Terrestrial planets are small, rocky planets like Earth, while gas giants are large, gaseous planets like Jupiter. Ice giants, like Uranus and Neptune, are similar to gas giants but contain more ice and less gas. Dwarf planets are small, spherical objects that do not have enough mass to clear their orbit of other objects.

How do scientists study star-planet systems?

Scientists use a variety of methods to study star-planet systems, including direct imaging, radial velocity, and transit photometry. Direct imaging involves taking pictures of the system using powerful telescopes. Radial velocity measures the tiny wobbles of a star caused by the gravitational pull of its orbiting planets. Transit photometry looks for dips in a star's brightness as a planet passes in front of it.

What can we learn from studying star-planet systems?

Studying star-planet systems can help us better understand the formation and evolution of our own solar system. It can also provide insight into the conditions necessary for life to exist on other planets. Additionally, studying these systems can help us improve our understanding of planetary dynamics and the effects of stellar radiation on planets.

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