Is the Sun's wobble more likely to be due to Jupiter or Earth?

Question

Suppose that an observer at Alpha Centauri could detect a wobble in the Sun due to the motions of its planets. Is the Sun's wobble more likely to be due to Jupiter or Earth? Explain answer.

Answer 1

Jupiter of course, because its 1000's of times more massive.

Answer 2

The wobble comes from the fact that the star has to orbit around the common centre of mass (Star + planet) in the same time as it takes the planet to go around the star.
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The centre of mass (barycentre) can be found with:

D = (m*d)/(M+m)

D is the distance of the barycentre from the centre of the Sun.
d is the distance of the planet from the Sun.
both D and d will be in the same units (e.g., km, mi, AU)

M is the mass of the Sun and m is the mass of the planet (both have to be in the same units).

Let us use AU as the distance unit and the mass of Earth as the mass unit.

For Earth (d = 1, m = 1, M = 332,946)
D = (1*1) / 332947 = 0.000003 AU = 450 km (approx.)
The sun must "orbit" this 450 km radius in one year.

circumference = 2831 km = 2,831,000 m
time = 31,557,000 seconds (approx)
wobble speed = 2,831,000 m / 31,557,000 s =
0.089 m/s
9 cm per second.

For Jupiter (d= 5.2, m=317.8, M = 332,946)
D = (5.2*317.8) / 333,263.8 = 0.00496 AU
D = 743,810 km (this places it just OUTSIDE the sun's surface).

Circumference = 4,673,500 km
Period = 11.86 * 31,557,000 s
Wobble speed = 0.0125 km/s = 12.5 m/s

Wobble cause by Jupiter is 140 times more than wobble caused by Earth

Answer 3

Let's see: Jupiter is about 318 times a massive as the Earth but orbits about 5.5x further out.

F = G*M1*M2/r**2

The difference in mass means it gives Jupiter roughly 300x more gravitational effect than the Earth, but the difference in radius means it has about 1/30 the effect. The net result is that Jupiter exerts about 10x the gravitational effect than the Earth on the Sun.

So the winner is...

Answer 4

Gravitational Force = (G)(m1)(m2)/(r^2)

In your question, you only need to compare earth and Jupiter's masses, and their distances from the sun.

Jupiter's mass is about 318 times more than earth.
Juptier's distance from the sun is about 5.2 times farther away.

Earth: 1/(1)^2 = 1
Jupiter: 318/(5.2)^2 = 11.8

So, Juptier has a stronger gravitational effect on the sun than earth does, because it is so much more massive than earth. Plus it's really not all that far away.

Note, if you do this for Saturn, you'll see that Saturn and earth exert almost the same force on the sun.

Answer 5

well all of the planets affect the wobble some but since jupiter has more mass it would affect it more.