Finding mass of a central body: centripetal forces and orbiting bodies

In summary: This is why I do not use powers in this problem.)In summary, the mass of Epsilon Eridani in stellar masses (Ms) can be calculated using the length of Earth's semi-major axis (1AU = 1.49 x 10^11 m) and the orbital period of its planet (6.54 years). The final answer is approximately 0.919 solar masses, which is within 2% of the accepted value. However, using more accurate values for 1 AU and solar mass can yield a closer result. It is important to use precise values and avoid raising approximate numbers to powers in order to maintain accuracy in calculations.
  • #1
Omnistegan
3
0

Homework Statement



One stellar mass is defined as the mass of our sun (1Ms = 1.99 x 1030 kg). ONe astronomical unit is defined as the length of Earth's semi-major axis (1AU = 1.49 x 1011 m). The star Epsilon Eridani in the constellation Eridanus has a planet (discovered in 2000) orbiting it that has a semi-major axis of 3.39 AU. The orbital period of the planet is 6.54 years. Based on this information, determine the mass of Epsilon Eridani in stellar masses (Ms).

Homework Equations



[tex]F_{c} = F_{g}[/tex]
[tex]F_{c} = \frac{4\pi^2r}{T^2}[/tex]
[tex]F_{g} = \frac{Gm_{1}m_{2}}{r^2}[/tex]

The Attempt at a Solution



mp is the mass of the planet, me is the mass of Epsilon Eridani
[tex]\frac{4\pi^2m_{p}r}{T^2} = \frac{Gm_{p}m_{e}}{r^2}[/tex]
Solve for me, the mp's cancel
[tex]m_{e} = \frac{4\pi^2r^3}{T^2G} = \frac{4\pi^2\left(3.39 \times 1.49\times 10^{11}\right)^3}{\left(6.54 \times 365 \times 24 \times 3600\right)\left(6.67\times 10^{-11}\right)} = 1.79 \times 10^{30}[/tex]
now divide that answer by kg in a Stellar Mass
[tex]\frac{1.79 \times 10^{30}}{1.99 \times 10^{30}} = 0.901M_{s}[/tex]

Apparently the correct answer is 0.903Ms. I did have a chance to clarify with my teacher that 365x24x3600 is what he expected us to use for seconds.
Any help is appreciated!
 
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  • #2
And you need to calculate this exactly?

Within .2% of the answer is not acceptable, even with the precision given in the problem?
 
  • #3
I agree with LowlyPion here.

BTW, if you want to be precise, you should have used 1 year = 365.242 days, 1 solar mass = 1.98892×1030 kilogram, 1 AU = 1.49598×1011 meters.

However, this will give 0.911 solar masses as the final answer.

If you simply substitute 1.50e11 meters for the length of 1 AU for your value of 1.49e11 meters yields a final answer of 0.919 solar masses. Note well: 1.50e11 meters is a better 3-digit value for the length of 1 AU than is 1.49e11 meters.So what is going on?

(1) Just because you only know some values to 3 digits does not mean you should truncate everything to 3 digits. It is far better to represent physical constants to their full accuracy and truncate the final result to the expected accuracy (e.g., three digits in this case).

(2) A long sequence of products and ratios involving approximate numbers can (will) reduce the accuracy of your final result.

(3)Raising approximate numbers to powers can (will) reduce the accuracy of your final result.
 

1. How is the mass of a central body determined in a system of orbiting bodies?

In a system of orbiting bodies, the mass of the central body can be determined by measuring the orbital period and distance of an orbiting object using Kepler's laws of planetary motion. By plugging these values into the equation for centripetal force, the mass of the central body can be calculated.

2. What is the relationship between centripetal force and the mass of the central body?

The centripetal force acting on an orbiting body is directly proportional to the mass of the central body. This means that as the mass of the central body increases, the centripetal force also increases, resulting in a faster orbital speed for the orbiting body.

3. How does the distance between the central body and an orbiting body affect the centripetal force?

The centripetal force is inversely proportional to the square of the distance between the central body and the orbiting body. This means that as the distance increases, the centripetal force decreases, resulting in a slower orbital speed for the orbiting body.

4. Can the mass of the central body be determined if only one orbiting body is present?

Yes, the mass of the central body can still be determined if only one orbiting body is present. By measuring the orbital period and distance of the orbiting body, and using the equation for centripetal force, the mass of the central body can be calculated.

5. Is the mass of the central body always constant in a system of orbiting bodies?

In most cases, the mass of the central body is considered to be constant in a system of orbiting bodies. However, in some cases where there are multiple orbiting bodies, the gravitational interactions between them may slightly affect the mass of the central body.

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