Why do different formulae give me different answers for mass of galaxy?

[anyaxo]

I am trying to calculate the mass of the galaxy within the orbit of the sun.

M = Mass of the galaxy = ?
r = ‘Distance of sun from galactic centre = 7.6 kpc’ = 2.3 x 1017 km = 2.3 x 1020 m (2 sf)
G = Gravitational constant = 6.67 x 10-11 Nm2kg-2
v = ‘Radial velocity of the sun = 220,000 ms-2’

Since the Sun’s orbit around the galactic centre is nearly circular, it undergoes circular motion and experiences a centripetal force:

〖mv〗^2/r = - GMm/r^2

v^2 = GM/r
M= ((220,000)^2 (2.3 ×〖10〗^20))/(6.67 ×〖10〗^(-11) )
=1.7 × 〖10〗^41 kg (2 significant figures)
Using Newton's form of Kepler's Third Law..

r = ‘distance of sun from galactic centre = 1.6 x 109 AU (Astronomical Units)’ 13
T = ‘orbital period of the sun = 2.4 x 108 Julian Years’
M = Mass of galaxy = ?

M= r^3/T^2

M= 〖(1.6 × 〖10〗^9)〗^3/〖(2.4 ×〖10〗^8)〗^2

M=7.1 × 〖10〗^10 M_ʘ

This proves that if every star has approximately the mass of the sun, there are approximately 7.1 x 10^10 stars in the Milky Way.

7.1 x 10^10 Mʘ = 1.4 x 10^41 kg (2 significant figures)
Finally, by using the original form of Kepler’s Third Law we can also obtain the mass of a galaxy.

T = ‘orbital period of the sun = 2.4 x 108 Julian Years’ 15 = 7.57 x 1015 seconds
r = ‘distance of sun from galactic centre = 2.3 x 1017 km’ 13= 2.3 x 1020 m
G = Gravitational constant = 6.67 x 10-11 Nm2kg-2

T^2=(〖4π〗^2/GM)r^3

M= (4π^2 r^3)/(GT^2 )

M= (4 × π^2 ×〖(2.3 × 〖10〗^20)〗^3)/((6.67 × 〖10〗^(-11) ) 〖(7.57 × 〖10〗^15)〗^2 )

M = 1.3 x 10^41 kg (2 significant figures)

Why are my three answers (all in bold) so different? and which one is correct and why? I need to explain this for my coursework!

 

[Delphi51]

Using your values for R and T, I get v = 2πr/T = 1.91 x 10^5 m/s, which is not the same as the 220,000 m/s you use in the first calc.