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

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To calculate the mass of the Sun using orbital mechanics, the equation Ms = 4(pi)^2R^3/(G)(T^2) is applied, where R is the Earth's orbital radius and T is the orbital period in seconds. The user attempted the calculation with R = 1.5X10^11 m and T = 365.25 days converted to seconds. The resulting mass was incorrectly calculated as 6.434X10^37 kg, likely due to errors in exponent handling or the squared term for T. A reminder was provided to ensure T is squared in the equation, indicating a potential source of error. Accurate calculations are crucial for deriving the correct mass of the Sun.
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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.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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