Escape Velocity of the Milky Way

In summary, the Milky Way has a flat rotation curve in its Solar neighborhood, with a constant velocity Vc. This implies a mass density profile of ρ(r) ~ r^-2. The velocity of escape from the galaxy at any radius r<R is given by Ve^2= 2Vc^2(1+ln R/r) and can be calculated by finding the change in potential energy and the required kinetic energy.
  • #1

Homework Statement



In the Solar neighborhood, the Milky Way has a flat rotation curve, with V(r)= Vc where Vc is a constant, implying a mass desnity profile ρ(r) ~ r^-2

Assume there is a cutoff radius R beyond where the mass density is zero. Prove that the velocity of escape from the galaxy from any radius r<R is:

Ve^2= 2Vc^2(1+ln R/r)


Homework Equations



Integral needs to be done in two parts


The Attempt at a Solution



The 1/2 mv^2 provides the energy needed to do the work of moving the mass m against the force of gravity from a radius r to infinity.
I believe the integral needs to be evaluated at both r and R, however, I do not know what equation to integrate because integrating 1/2 mv^2 doesn't seem like it will yield the above equation.
 
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  • #2
A body orbiting a central mass M in a circular orbit at radial distance r has orbital velocity
[tex] V_c = \sqrt{\frac{G M}{r}} [/tex]
For the galaxy then, if the velocity profile is flat then Vc is a constant from r to R, and you can obtain an effective mass w.r.t. radius by solving the above for M and calling it M(r).

With M(r) you are in a position to determine the change in PE for an object taken from r to R.

Next, consider that at distance R the circular orbital velocity is still Vc, and that since all the effective mass is "below" R, it will behave as a point mass at the center and the escape velocity there must be [itex]\sqrt{2}V_c[/itex]. That gives you the required KE at radius R.

For escape, the starting KE will be the required final KE plus the loss in PE. Find the escape velocity from the starting KE.
 

What is the Escape Velocity of the Milky Way?

The escape velocity of the Milky Way refers to the minimum speed that an object needs to achieve in order to escape the gravitational pull of the galaxy and move into intergalactic space.

How is the Escape Velocity of the Milky Way calculated?

The escape velocity of the Milky Way can be calculated using Newton's law of universal gravitation, which takes into account the mass and distance of the galaxy. It can also be estimated by measuring the velocities of stars and gas particles at different distances from the center of the galaxy.

What is the current estimate for the Escape Velocity of the Milky Way?

The current estimate for the escape velocity of the Milky Way is approximately 550 km/s. However, this value can vary depending on the location within the galaxy and the method used for calculation.

How does the Escape Velocity of the Milky Way compare to other galaxies?

The escape velocity of the Milky Way is relatively low compared to other galaxies of similar size. This is because the Milky Way has a relatively low mass compared to its size, resulting in weaker gravitational pull.

What implications does the Escape Velocity of the Milky Way have for space exploration?

The escape velocity of the Milky Way makes it difficult for objects to leave the galaxy, which can be a challenge for space exploration. However, it also provides a natural protection for the galaxy from external objects and may also play a role in the formation and evolution of the Milky Way.

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