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

[eraemia]

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!

 

[genneth]

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?

 

[eraemia]

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!

 

[Saladsamurai]

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