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Chapter 2 - Measuring the Mass of the Sun |
This chapter of the video
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shows how we can use Newton's theory of universal gravitation to calculate the mass of the Sun from the orbit
of any planet.
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The method for measuring mass discussed in this chapter of the video is commonly known as the Dynamical Method. However, we use the term Orbital Method to emphasize its connection to the orbital speeds of stars.
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Eliptical Orbits
In calculating the mass of the Sun, we model Jupiter's orbit around the Sun as being circular, as in Figure 8. Although its orbit is actually elliptical, this fact makes little difference to the result. Jupiter's orbit is only slightly elliptical and the difference between the masses calculated assuming circular and elliptical orbits is less than 0.01%.
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Difference Between Circular and Eliptical Orbits
As planets orbit the Sun in elliptical orbits, as in Figure 9, we can derive the Sun's mass from Kepler's third law of motion:
Where T is the planet's period, a is the semi-major axis of the orbit, and MS is the mass of the Sun. Rearranging Equation 2.1 to solve for MS , we obtain:
Jupiter's orbit is very close to circular. It is so close that the difference in the result for MS we obtain by modelling Jupiter's orbit as being circular instead of elliptical is just 0.0064%. This corresponds to the difference obtained by using the equation:
Instead of:
Thus, using a circular orbit for Jupiter leads to a highly accurate result for the mass of the Sun. Furthermore, using such an orbit for any other planet also yields an accurate result for this mass.
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