Angular Momentum of a Body Constantly Losing Mass My friend has a question:  assume that a body such as a star loses mass as it orbits the galaxy. If there is conservation of angular momentum, how is this explained? What happens to its orbital velocity and orbital radius? Thanks.
 A: Suppose a star puffs off some of its mass as a planetary nebula. This is a common mechanism by which stars lose mass. The mass blown off the star has an angular momentum because it too is orbiting around the centre of the galaxy, and if you add up the angular momentum of the star and of the matter it has ejected then the total angular momentum is conserved.
In a planetary nebula the matter is ejected roughly evenly so the velocity and orbital radius of the star doesn't change. If the matter were ejected preferentially in one direction then it would propel the star in the other direction and this would change the orbit of the star. This is no different to the way rockets change their orbits by firing their motors - a rocket motor just ejects matter though in a rather more controlled way than a star ejecting matter.
A: In a system in which external torque is $0$, total angular momentum is conserved. This means that, if you neglect the influence of other planets and stars in the galaxy, the total momentum (spin+orbital!) of your star will be conserved. This includes the angular momentum of eventual expelled mass, so there is no contradiction. 
A: I know that this is an old question, but I find the three answers thus far to be unsatisfying.  Not to disagree with any of the posted answers, just to provide a little more detail in language which is more plain.
To answer your question directly, and to echo what has already been said, it's orbital velocity and orbital radius will remain unchanged.  The star itself will lose momentum because some of that momentum is being carried away in the mass which has been "lost".  Because the mass isn't really "lost".  Total angular momentum is conserved because the total mass of the system remains unchanged.
May I change your question?  Galactic radii are so large as to consider the travel of the star to be translational.  Technically rotational on a huge time scale, but generally translational over a couple of million years.
When considering conservation of angular momentum it is often helpful to consider the problem as translational (linear) rather than rotational.  Consider an object of given mass traveling through space at a given velocity.  Part way through its travel it separates in half, into two objects of equal mass.  Does the original object move faster or slower?  The answer, of course, is that both objects continue to move at the same velocity, both taking half the momentum (because they each took half the mass) with them.
Getting back to a rotational example, let's consider the example of a figure skater holding two bowling balls.  The figure skater enters into a spin, draws his arms in close, is spinning incredibly fast, then lets go of the bowling balls.
In order to conserve momentum does the figure skater spin faster or slower after releasing the bowling balls?
Neither; his angular velocity remains unchanged.  Momentum is "conserved" in that the component of the momentum being contributed by the bowling balls gets carried away in the form of translational momentum.  The moving bowling balls still have momentum even though they are no longer part of the spinning skater.  If the result of releasing the balls had caused the skater to began to spin faster to "conserve" his angular momentum, then we've just magically created momentum out of nothing -- a magic momentum machine.  But if you take the total momentum of the system -- the skater and the two released bowling balls -- and add up all their momenta you will find that the total momentum of all of the objects adds up to be the same as the original momentum they had when they were all attached.
Hope that helps.

My friend has a question: assume that a body such as a star loses mass
  as it orbits the galaxy. If there is conservation of angular momentum,
  how is this explained? What happens to its orbital velocity and
  orbital radius? Thanks.

A: Well, the angular momentum is conserved when net torque on a body(or a system) is zero.
Now in your case, the torque produced by the gravitational force between the galaxy and the star is zero as the line of force passes through the axis of rotation.
For simplicity, i am neglecting any effect produced by other celestial objects of the universe.
Now, if the mass separates out tangentially from the star with the same speed, then it will apply no force on the star and as a result the speed of the star will remain unaltered. and if the speed is same, the star and the mass  will revolve in the same orbit with same speed.(because the radius of orbit is given by $r=GM/V^2$ where M is the mass of galaxy) However if due to internal forces, the mass separates out with some different speed, then the internal force will change the linear momentum of the star(note that the linear momentum of the star+mass will be conserved for that time when it separates.).
As the Momentum changes, the speed will change and as a result the orbit too will change.
However, in each case the angular  momentum of the star+mass will be conserved. Also, if your question is concerned about the angular momentum of the star only(not the mass), then it will not be conserved until the mass separates out with no speed.
