Question

One method of detecting extrasolar planets involves looking for indirect evidence of a planet in the form of wobbling of its star about the star-planet system's center of mass. Assuming that the Solar System consisted mainly of the Sun and Jupiter, how much would the Sun wobble? That is, what back-and- forth distance would it move due to its rotation about the center of mass of the Sun-Jupiter system? How far from the center of the Sun is that center of mass?

Short answer

Answer: The Sun would wobble back and forth by 7.54 * 10^8 meters.

Step-by-step solution

Find the mass of the Sun and Jupiter

For the masses of the Sun and Jupiter, we have: Mass of the Sun (mass1) = 1.989 * 10^30 kg Mass of Jupiter (mass2) = 1.898 * 10^27 kg

Set the distance between the Sun and Jupiter

As we are interested in the distance the Sun would move, let's consider Sun's position as the origin point: Position of the Sun (position1) = 0 Distance between the Sun and Jupiter (position2) = 7.78 * 10^11 m (This is the average distance)

Calculate the center of mass

Now, we will calculate the center of mass of the Sun-Jupiter system using the given formula: center_of_mass = (mass1 * position1 + mass2 * position2) / (mass1 + mass2) center_of_mass = (1.989 * 10^30 kg * 0 + 1.898 * 10^27 kg * 7.78 * 10^11 m) / (1.989 * 10^30 kg + 1.898 * 10^27 kg)

Simplify the equation and solve for the center of mass

After calculating the center of mass, we get: center_of_mass = 7.54 * 10^8 m

Determine the amount of wobble in the Sun

The amount of wobble in the Sun's movement is the same as the distance between its center and the center of mass of the Sun-Jupiter system. Therefore, the Sun would wobble back and forth by 7.54 * 10^8 m.

Conclusion

In conclusion, by analyzing the two-body system of the Sun and Jupiter, we calculated that the Sun would have a wobble with a back-and-forth distance of 7.54 * 10^8 m, and the center of mass of the Sun-Jupiter system would be 7.54 * 10^8 m away from the center of the Sun.

Key concepts

Center of Mass

The center of mass of a system represents the average position of all the mass in a system. In simpler terms, it's the balance point of the system. If you could place a finger under the system at this exact point, it would balance perfectly, assuming gravity acted uniformly on the system. For celestial bodies, the center of mass is crucial to understand how they orbit each other.

Returning to our Solar System example, when dealing with the Sun and Jupiter, we're essentially looking for the point in space where the two objects' gravities balance each other out. This spot is not necessarily within either body, especially if one body is much more massive than the other, as is the case with the Sun and Jupiter. By calculating the center of mass, we can predict the motion of the two bodies around this point, which is pivotal in detecting the 'wobble' indicative of an extrasolar planet.

Two-Body System

A two-body system in astrophysics refers to two celestial bodies which are gravitationally bound to each other and revolve around a common center of mass. The Sun and Jupiter pairing is an excellent example of a two-body system. This system can be simplified: as the Sun is so much heavier than Jupiter, you might think that Jupiter orbits the Sun as if it were stationary.

However, what actually happens is both the Sun and Jupiter orbit the center of mass of the system. This common center of mass is closer to the larger body, which in this case is the Sun. This sort of system helps us understand the orbital mechanics and provides insights into the movement caused by mutual gravitational attraction. Recognizing this mutual orbit is integral to spotting the signs of extrasolar planets.

Gravitational Wobble

The term 'gravitational wobble' refers to the slight movement of a star caused by the gravitational pull of an orbiting planet. When a planet orbits a star, its gravity pulls on the star causing it to move in a small orbit around the center of mass of the system, as we observed with the Sun and Jupiter. This movement is often referred to as a 'wobble'.

While the wobble caused by a giant planet like Jupiter is detectable with current technology, even smaller planets can cause a measurable wobble if instruments are sensitive enough. Detecting this wobble is a cornerstone method, known as the radial velocity method, employed in the search for extrasolar planets. The ability to measure this wobble provides invaluable indirect evidence of a planet's presence, especially when the planet itself is too dim to be observed directly.

Celestial Mechanics

Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of celestial bodies. This field underpins our exercise as it provides the formulas and theories needed to calculate the center of mass, orbits, and the effects of bodies on each other in a two-body system.

In the context of detecting extrasolar planets, celestial mechanics provides us with the knowledge to interpret the gravitational wobble. Utilizing the laws of physics, we can deduce the presence, mass, and orbit of an unseen planet based on the precise movements of its star. It's through celestial mechanics that we understand how stars and planets interact over vast distances, enabling us to unveil the existence of distant worlds orbiting other suns.