To find the gravitational force that a galaxy, modeled as a gas cloud with a specific mass density, exerts on a point mass M located at a distance r from the center, we start by determining the total mass of the galaxy within a radius r. The mass density is given by P(r) = Po exp(-r/n), where Po is a constant and n is a scale length. The gravitational force can then be derived from the mass enclosed within that radius.
Calculating the Total Mass Enclosed
The total mass enclosed within a radius r can be found by integrating the mass density over the volume of a sphere of radius r. The volume element in spherical coordinates is dV = 4πr² dr. Thus, the total mass mtot within radius r is given by:
mtot(r) = ∫ P(r) dV = ∫ P(r) 4πr² dr
Substituting the expression for P(r):
mtot(r) = ∫ (Po exp(-r/n)) 4πr² dr
This integral can be simplified to:
mtot(r) = 4πPo ∫ r² exp(-r/n) dr
Integrating by Parts
To solve the integral ∫ r² exp(-r/n) dr, we can use integration by parts. Let:
- u = r² → du = 2r dr
- dv = exp(-r/n) dr → v = -n exp(-r/n)
Applying integration by parts:
∫ r² exp(-r/n) dr = uv - ∫ v du
Substituting the values:
∫ r² exp(-r/n) dr = -n r² exp(-r/n) - ∫ (-n)(2r) exp(-r/n) dr
This leads to:
∫ r² exp(-r/n) dr = -n r² exp(-r/n) + 2n ∫ r exp(-r/n) dr
We need to integrate ∫ r exp(-r/n) dr again by parts. Let:
- u = r → du = dr
- dv = exp(-r/n) dr → v = -n exp(-r/n)
Applying integration by parts again gives:
∫ r exp(-r/n) dr = -n r exp(-r/n) - ∫ (-n) exp(-r/n) dr
Thus, we have:
∫ r exp(-r/n) dr = -n r exp(-r/n) + n² exp(-r/n)
Now substituting back into our earlier expression:
∫ r² exp(-r/n) dr = -n r² exp(-r/n) + 2n(-n r exp(-r/n) + n² exp(-r/n))
Combining these results will yield the total mass mtot(r).
Finding the Gravitational Force
The gravitational force F exerted by the galaxy on the point mass M at distance r can be derived from Newton's law of gravitation:
F = G (M * mtot(r)) / r²
Substituting mtot(r) into this equation gives:
F(r) = G (M * (4πPo ∫ r² exp(-r/n) dr)) / r²
This expression will yield the gravitational force as a function of r, incorporating the total mass of the galaxy.
Graphing the Force Function
To visualize how the gravitational force F varies with distance r, we can sketch a graph. Generally, the force will start strong when r is small (as the mass enclosed is significant) and will decrease as r increases, reflecting the diminishing influence of the galaxy's mass at larger distances.
In summary, the gravitational force F exerted by the galaxy on the point mass M can be expressed in terms of the total mass enclosed within radius r, and the integration process involves careful application of integration by parts to handle the exponential decay of the mass density. The resulting force function will typically show a decrease with increasing distance, characteristic of gravitational interactions.
