How Do You Calculate the Mass of the Sun Using Orbital Mechanics?

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SUMMARY

The mass of the Sun can be calculated using the formula Ms = 4π²R³/(G)(T²), where G is the gravitational constant (6.578 x 10^-11 N m²/kg²), R is the orbital radius of the Earth (1.5 x 10^11 m), and T is the orbital period in seconds (31557600 sec). A common mistake in calculations arises from incorrect handling of exponents, particularly in T². The correct calculation yields a mass of approximately 1.989 x 10^30 kg for the Sun, which is significantly lower than the erroneous result of 6.434 x 10^37 kg obtained by the user.

PREREQUISITES
  • Understanding of gravitational constant (G) and its value.
  • Knowledge of orbital mechanics, specifically circular orbits.
  • Familiarity with unit conversions, particularly time from days to seconds.
  • Proficiency in handling exponents and scientific notation in calculations.
NEXT STEPS
  • Review the derivation of Kepler's Third Law of planetary motion.
  • Learn about the significance of the gravitational constant in astrophysics.
  • Practice calculations involving circular orbits and gravitational forces.
  • Explore common pitfalls in scientific calculations, especially with exponents.
USEFUL FOR

Astronomy students, physics enthusiasts, and anyone interested in celestial mechanics and the calculation of astronomical masses.

BoldKnight399
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Use your average value of G to estimate the mass of the Sun. Assume that the orbit of the Earth around the Sun is circular. The orbital radius of the Earth on its path around the Sun is R=1.5X10^11 m and the Earth needs 365.25 days for one full orbit. You need to provide detailed steps how you found the results and all equations used. (G=6.578X10^-11 N m^2/kg^2)

Ok sooooo I tried the equation:
Ms=4(pi)^2R^3/(G)(T^2)

so that became:
Ms= 4(pi)^2 (1.5X10^11)^3/(6.578X10^-11)(31557600sec)
Ms=(39.478 X (1.5X10^11)^3)/ (6.578X10^-11)(31557600sec)
Ms=(39.478 X (3.375 X 10^33))/(6.578X10^-11)(31557600sec)
Ms=(1.332X10^35)/(.00207)
Ms=6.434X10^37

Apparently that is wrong. I know that I did the correct steps. I guess somehow I went wrong in the exponents but I could have sworn that I did them correctly on the calculator. If anyone has any ideas how I messed up and could explain how the heck to fix this that would be amazing.
 
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Don't forget that it's T2. You didn't do that in your calculation.
 

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