a) Calculate the reflex motion of the Sun (its radial velocity as it orbits around the center of mass) due to the presence of Jupiter. Jupiter’s orbital period is 11.86 yr, average distance from the Sun is 5.2 AU, and mass is 0.000955 M⊙. You may assume that Jupiter’s orbit is circular.

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1.
a) Calculate the reflex motion of the Sun (its radial velocity as it orbits around the center of mass) due to the presence of Jupiter. Jupiter’s orbital period is 11.86 yr, average distance from the Sun is 5.2 AU, and mass is 0.000955 M⊙. You may assume that Jupiter’s orbit is circular.

b) What is your answer in (a) equivalent to in terms of another common-day speed that you know of (a runner, walker, car, a cheetah, a bird flying, etc.).

c) Repeat the calculation in (a) for Saturn and compare your result to the one you got in (a), commenting on what you think an alien civilization in a nearby star system would conclude about our own solar system if they were making radial velocity measurements of the Sun.

EXPERT ANSWER

Answer-1)

Part-a)

Given:

The orbital period of Jupiter = 11.86 yr = 3.74017x 108 s.

[1 year = 365 x 24 x 60 x 60 = 3.154 x 107 s]

The average distance of Jupiter from the Sun = 5.2 AU = 7.779 x 1011 m.

[1AU = 1.496×1011 m]

The mass of the Jupiter = 0.000955 M⊙

[where M⊙ is the mass of the sun. ]

To calculate: The radial velocity of the sun (i.e the reflex motion of the Sun) due to the presence of Jupiter.

Solve-a)

\rightarrow First we will calculate the Orbital velocity of the Jupiter. To do so we will use the following Formulae-

Orbital Velocity (m/s) = 2πR/T

where R is the average radius of the orbit (i.e The average distance of Jupiter from the Sun ) in meters and T is the orbital period in seconds.

Substituting the values of radius of the orbit and the orbital period in the above relation we get-

Orbital Velocity (m/s) = [2 x 3.14 x 7.779 x 1011] / [3.74017x 108 ]

Orbital Velocity (m/s) = 13061.47.

So the Orbital velocity of the Jupiter = 13061.47 m/s.

\rightarrow Now we will calculate the reflex motion of the Sun.

Now we know that the when Jupiter revolve around the sun, the momentum of both the bodies are conserved. So we can use the following relation to calculate the reflex motion of the Sun (i.e reflex motion of the Sun )

M_{s} \times V_{s} = M_{J} \times V_{J}

Where Ms is the mass of the sun , Vs is the velocity of the sun, MJ is the Mass of the Jupiter and VJ is the velocity of the Jupiter.

Substituting the values of Orbital velocity of the Jupiter, mass of the Jupiter in the above relation we get-

M_{s} \times V_{s} = 0.000955 M_{s} \times 13061.47 m/s

Note: Ms is the mass of the sun

V_{s} = 0.000955 \times 13061.47 m/s
V_{s} = 12.5 m/s

Answer = The radial velocity of the sun (i.e the reflex motion of the Sun) due to the presence of Jupiter = 12.5 m/s.

Part-b)

Answer-b) we know that the speed of the running cheetah is around 25 m/s. Now expressing the above obtained answer in the common-day speed (i.e speed of the running cheetah) is-

The radial velocity of the sun (i.e the reflex motion of the Sun) due to the presence of Jupiter is half of the speed of the running cheetah.

Part-c)

To calculate: The radial velocity of the sun (i.e the reflex motion of the Sun) due to the presence of Saturn.

Before further calculation we will need the following data which can be easily found in the textbook or using the internet.

The orbital period of Saturn = 29 yr = 9.145 x 108 s.

[1 year = 365 x 24 x 60 x 60 = 3.154 x 107 s]

The average distance of Saturn from the Sun = 9.5 AU = 1.421 x 1012 m.

[1AU = 1.496×1011 m]

The mass of the Saturn = 0.0002857 M☉

[where M⊙ is the mass of the sun. ]

Solve-c)

\rightarrow First we will calculate the Orbital velocity of the Saturn. To do so we will use the following Formulae-

Orbital Velocity (m/s) = 2πR/T

where R is the average radius of the orbit (i.e The average distance of Saturn from the Sun ) in meters and T is the orbital period in seconds.

Substituting the values of radius of the orbit and the orbital period in the above relation we get-

Orbital Velocity (m/s) = [2 x 3.14 x 1.421 x 1012 ] / [9.145 x 108 ]

Orbital Velocity (m/s) = 9758.20.

So the Orbital velocity of the Saturn = 9758.20 m/s.

\rightarrow Now we will calculate the reflex motion of the Sun.

Now we know that the when Saturn revolve around the sun, the momentum of both the bodies are conserved. So we can use the following relation to calculate the reflex motion of the Sun (i.e reflex motion of the Sun )

M_{s} \times V_{s} = M_{sat} \times V_{sat}

Where Ms is the mass of the sun , Vs is the velocity of the sun, Msat is the Mass of the Saturn and Vsat is the velocity of the Saturn.

Substituting the values of Orbital velocity of the Saturn, mass of the Saturn in the above relation we get-

M_{s} \times V_{s} = 0.0002857 M_{s} \times 9758.20 m/s

Note: Ms is the mass of the sun.

V_{s} = 2.78 m/s

Answer = The radial velocity of the sun (i.e the reflex motion of the Sun) due to the presence of Saturn = 2.78 m/s.

\rightarrow we can see that the the reflex motion of the Sun due to the presence of Jupiter is much larger compare to the reflex motion of the Sun due to the presence of Saturn.

\rightarrow Alien civilization in a nearby star system would conclude that there is an planet revolving around the sun by making radial velocity measurements of the Sun. This is som because the radial velocity of star arises due to the revolving planet as it is seen in the above problems.