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# Orbital Radius & Detection Techniques of Extrasolar Planets

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Please show all calculations.

I only need help with part (a)i and ii of the question.

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## SOLUTION This solution is FREE courtesy of BrainMass!

I'll start by letting you know the key things you have to understand to address this question appropriately:

1. In a system like that of stars and planets, there is always a center of mass. This center of mass is always closer to the more massive body, and in (almost) all cases, the star in a system is more massive than the planets in the same system, the result being that the center of the mass for a star-planet system will be closer to the star than to the planet.

2. The star and planet both orbit their common center of mass in their different orbits; since the center of mass is always closer to the star than to the planet, the star's orbit is always shorter than the planet's, the result being that we hardly notice the star's motion as compared to the planet's. [Most people usually neglect the star's motion in a system especially if the star is much more massive than the planet, it will only appear the planet is the one orbiting the stationary star (like we always say that the earth and other planets orbit the sun), but this is not exactly true; the sun is also in orbital motion about the center of mass of the solar system just that this center of mass is very close to the sun due to its massiveness, and so the sun's orbit is also really too short compared to those of the planets, and we tend to ignore this motion]

3. Due to the brightness of the stars and their great distances away from us, it is impossible to directly image planets in their systems from earth with the technologies we have today, especially for really far systems like the 1000 pc mentioned in this question. [I have just indirectly said that Direct Imaging of distant extra-solar planets is not possible with our today's technology (this kind of answers the (ii) part of your problem].

Now we can begin answering your questions.

SOLUTION:
(i) To answer the (i) part of the question, you will need the formula below from the law of conservation of momentum;

(mass of star) * (distance of star from the system's center of mass) = (mass of planet) * (distance of planet from the system's center of mass)
------------Equation (1)

For planet X:
mass of star=1.9891 Ã— 1030 kg
mass of planet X=12*ME =12*5.972*1024 kg
distance of star from the system's center of mass= 2*104 km
And we are looking for the orbital radius of planet X (that is, its distance from the system's center of mass)

From equation (1), we get

distance of planet X from the system's center of mass = (mass of star) * (distance of star from system's center of mass)/(mass of planet X)

Therefore, distance of planet X from the system's center of mass = (1.9891 Ã— 1030 kg) * (2*104 km)/( 12*5.972*1024 kg)

= 5.5512*108 km

This distance is in km because the other distance used in the formula was also in km. To state the answer in astronomical units, you will have to divide by 149598000 since 1AU=149598000.

And that gives the orbital radius for planet X as (5.5512*108)/(149598000) AU = 3.7107 AU

Similarly for planet Y:
mass of star=1.9891 Ã— 1030 kg
mass of planet Y=150*ME =150*5.972*1024 kg
distance of star from the system's center of mass= 2*103 km
And we are looking for the orbital radius of planet Y (that is, its distance from the system's center of mass)

Again from equation (1), we get

distance of planet Y from the system's center of mass = (mass of star) * (distance of star from system's centre of mass)/(mass of planet Y)

Therefore, distance of planet Y from the system's center of mass = (1.9891 Ã— 1030 kg) * (2*103 km)/( 150*5.972*1024 kg)

= 4.4409*106 km

And to convert to astronomical units, we divide by 149598000 to get the orbital radius for planet Y as (4.4409*106)/(149598000) AU = 0.0297 AU

(ii) As I mentioned earlier, the systems are really far (about 1000 pc away), so we cannot consider the technique of Direct Imaging for detecting any of the planets X or Y [we can't pick any of their images with the most powerful telescopes we have today, we can rather pick their stars because of their brightness, and so the practice in astronomy today is often to use the following logic: if a star wobbles periodically, then there is likely to be a planet (or some planets) around it causing this wobbling, a couple of techniques are being employed to measure the extent of these star wobbling, and from them compute the existence of extra-solar planets around the stars].
(a) Astrometry will be an appropriate technique for detecting planet X. The method of Radial Velocity (also called Doppler Spectroscopy) will be appropriate for detecting planet Y. [see explanations below]
(b) Astrometry will be an inappropriate technique for detecting planet Y. Doppler Spectroscopy will be an inappropriate technique for detecting planet Y. [also see explanations below].
Astrometry involves measuring the star's changing position in the sky with time, and since planet X's system is orientated face-on (that is, perpendicular to our line of sight from earth), we can easily observe the star's motion over the sky. Precise measurements can help us get the extent of the star's motion and so compute the existence of planet X. This method will however be inappropriate for detecting planet Y since planet Y's system is orientated edge-on (that is, in our line of sight); the motion of planet Y's star will be more of towards and away from us, so there will be little or no change in its position in the sky from our view point, and as such this method cannot tell us whether or not this star is moving even if it really is. (see http://www.planetary.org/explore/topics/extrasolar/astrometry.html and http://en.wikipedia.org/wiki/Astrometry for wider reading on the technique).

In the technique of Doppler Spectroscopy, the star's motion toward or away from us is usually detected by measuring the Doppler shifts in frequency (or wavelength) of the radiation we get from it. In this way we are able to say if the star is moving towards or away from us and at what speed. It will be an appropriate method for detecting planet Y (whose system is orientated edge-on) since the motion of the star here is towards or away from us. On the contrary this method will be inappropriate for detecting planet X (whose system is orientated face-on) as its star is not wobbling in a direction towards or away from us, and as such the method will be unable to tell us if the star is moving at all. (see http://www.planetary.org/explore/topics/extrasolar_planets/extrasolar/radial_velocity.html and http://en.wikipedia.org/wiki/Doppler_spectroscopy for a wider reading).

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