The Sun, which is 2.2
1020 m from the center

of the Milky Way galaxy, revolves around that center once every

2.5
108 years. Assuming each star in the Galaxy has a mass equal

to the Sun’s mass of 2.0
1030 kg, the stars are distributed uniformly

in a sphere about the galactic center, and the Sun is at the

edge of that sphere, estimate the number of stars in the Galaxy.

To estimate the number of stars in the Milky Way galaxy, we can use some fundamental principles of physics, particularly those related to gravitational forces and the properties of spheres. Let's break this down step by step.

Understanding the Problem

We know the following key pieces of information:

• The distance from the Sun to the center of the Milky Way is approximately 2.2 × 10²⁰ meters.
• The Sun revolves around the galactic center once every 2.5 × 10⁸ years.
• The mass of each star is about 2.0 × 10³⁰ kg.

Using Gravitational Dynamics

The gravitational force acting on the Sun due to the mass of the stars in the galaxy can be described using Newton's law of gravitation. The centripetal force required to keep the Sun in its orbit is provided by this gravitational force.

Step 1: Calculate the Orbital Velocity

The orbital velocity (v) of the Sun can be calculated using the formula:

v = distance / time

Here, the distance is the circumference of the Sun's orbit, which can be calculated as:

C = 2πr, where r is the radius (2.2 × 10²⁰ m).

Thus, the circumference is:

C = 2π(2.2 × 10²⁰) ≈ 1.38 × 10²¹ m.

Now, we can find the orbital velocity:

v = (1.38 × 10²¹ m) / (2.5 × 10⁸ years × 3.15 × 10⁷ seconds/year) ≈ 1.75 × 10⁸ m/s.

Step 2: Apply Newton's Law of Gravitation

The gravitational force (F) acting on the Sun can be expressed as:

F = G(M * m) / r²,

where:

• G is the gravitational constant (6.674 × 10^-11 N(m/kg)²),
• M is the total mass of the stars within the radius r (which we need to find),
• m is the mass of the Sun (2.0 × 10³⁰ kg),
• r is the distance from the Sun to the center of the galaxy (2.2 × 10²⁰ m).

Step 3: Equate Gravitational Force and Centripetal Force

The centripetal force required to keep the Sun in its orbit is given by:

F_centripetal = m(v² / r).

Setting the gravitational force equal to the centripetal force gives us:

G(M * m) / r² = m(v² / r).

We can cancel m from both sides (assuming m ≠ 0) and rearrange to find M:

M = (v² * r) / G.

Step 4: Plugging in the Values

Now we can substitute the values we have:

M = ((1.75 × 10⁸)² * (2.2 × 10²⁰)) / (6.674 × 10^-11).

Calculating this gives:

M ≈ 1.4 × 10⁴¹ kg.

Estimating the Number of Stars

To find the number of stars (N) in the galaxy, we can divide the total mass by the mass of a single star:

N = M / mass of one star = (1.4 × 10⁴¹ kg) / (2.0 × 10³⁰ kg) ≈ 7 × 10¹⁰ stars.

Final Thoughts

Based on our calculations, we estimate that there are approximately 70 billion stars in the Milky Way galaxy. This number aligns with current astronomical estimates, which suggest that the Milky Way contains between 100 billion and 400 billion stars. The variation in estimates often arises from the challenges in observing distant stars and accounting for those that are not easily detectable.