Find the mass (in kg) of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is 1.18 x 10° m and its period is 1.12 days. How many Earth masses is this?
Find the mass (in kg) of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is 1.18 x 10° m and its period is 1.12 days. How many Earth masses is this?
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Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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![**Tutorial**
Find the mass (in kg) of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is \(1.18 \times 10^8\) m and its period is 1.12 days.
How many Earth masses is this?
**Part 1 of 3**
In order to find the mass, we need to combine the formulas for circular velocity and speed of the moon in its orbit.
\[
v = \frac{2\pi r}{P}
\]
\[
v = \sqrt{\frac{GM}{r}}
\]
Setting them equal gives:
\[
\sqrt{\frac{GM}{r}} = \frac{2\pi r}{P}
\]
Squaring both sides and solving for mass:
\[
\frac{GM}{r} = \left(\frac{2\pi r}{P}\right)^2
\]
Find the mass of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is \(1.18 \times 10^8\) m and its period is 1.12 days. Once you solve for the mass, does the exponent of the radius change? What is the exponent for \(r\)?
\[
M = \frac{4\pi^2}{G} \left(\frac{r_m}{P_s}\right)^2
\]
**Explanation:**
This exercise involves calculating the mass of Neptune using the orbital data of its moon, Proteus. The formulas provided relate to the circular motion of the moon in its orbit and involve equating the velocity formulas derived from the circumference of the orbit and gravitational forces. To find Neptune's mass (\(M\)), the simplified formula uses the radius of orbit (\(r_m\)) and the period (\(P_s\)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce65b68c-f7fa-43cb-8208-fb11f3ef12a3%2F059cb289-9d74-4237-b9c7-8461556e7785%2Fibvf7y9_processed.png&w=3840&q=75)
Transcribed Image Text:**Tutorial**
Find the mass (in kg) of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is \(1.18 \times 10^8\) m and its period is 1.12 days.
How many Earth masses is this?
**Part 1 of 3**
In order to find the mass, we need to combine the formulas for circular velocity and speed of the moon in its orbit.
\[
v = \frac{2\pi r}{P}
\]
\[
v = \sqrt{\frac{GM}{r}}
\]
Setting them equal gives:
\[
\sqrt{\frac{GM}{r}} = \frac{2\pi r}{P}
\]
Squaring both sides and solving for mass:
\[
\frac{GM}{r} = \left(\frac{2\pi r}{P}\right)^2
\]
Find the mass of Neptune from the orbital parameters of its moon, Proteus. The radius of Proteus's orbit about Neptune is \(1.18 \times 10^8\) m and its period is 1.12 days. Once you solve for the mass, does the exponent of the radius change? What is the exponent for \(r\)?
\[
M = \frac{4\pi^2}{G} \left(\frac{r_m}{P_s}\right)^2
\]
**Explanation:**
This exercise involves calculating the mass of Neptune using the orbital data of its moon, Proteus. The formulas provided relate to the circular motion of the moon in its orbit and involve equating the velocity formulas derived from the circumference of the orbit and gravitational forces. To find Neptune's mass (\(M\)), the simplified formula uses the radius of orbit (\(r_m\)) and the period (\(P_s\)).
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