The Sun orbits the center of the Galaxy in 225 million years at a distance of 26,000 light-years. Given that a 3 = (M1 + M2 ) × P 2 , where a is the semimajor axis and P is the orbital period, what is the mass of the Galaxy within the Sun's orbit?

 

a³ = (m₁ + m₂) P²

in this formula

a is the semimajor axis in AU,

P is the orbital period in Earth years, and

m is mass of object in relation to Sun's mass M?

Thus, we have to cenvert the given

1 ly = 9.46 x 10¹⁵ m

1 AU = 1.5 x 10¹¹ m

a = 26000 ly (9.46 x 10¹⁵ m / 1 ly )(1 Au / 1.5 x 10¹¹ m)

a = 1.64 x 10^9 Au

a³ = (m_sun + m_galaxy) P²

(a³ / P²) = (m_sun + m_galaxy)

since m_sun = 1 M? and is a very small fraction of m_galaxy the formula can be

m_galaxy = (a³ / P²)

m_galaxy = (1.64 x 10⁹)³ / (225 x 10⁶)²

m_galaxy = 8.78¹⁰ M?