Criterion for galaxy mergers, negative E? 
I'm writing about galaxy mergers and came across this section in a book describing criteria for a merger. I don't understand how there can be a negative $\hat{E}$ yet this is the region where mergers can occur, I would appreciate any explanation asap please!
 A: In dynamics, when you're looking at energies you have to define the zero-point: there isn't any particular, 'necessary' meaning of $E=0$.  The only thing that effects calculations is differences in energies.  It is often very convenient, and almost always the standard practice (when possible), to define $E=0$ as "unbound".  In other words, set $E=0$ to be the energy two gravitating particles would have if they were stationary (no kinetic energy), at an infinite separation (zero potential energy).
Kinetic energy, which makes particles less bound is positive, while potential energy which makes particles more bound is negative.  So it almost ends up being a tautology that only galaxies with negative energy can collide---because by definition those are the ones that are gravitationally "bound" to each-other.
Being bound (negative energy) is not enough however, because they may just orbit each other is a stable way.  For a stable orbit, they require sufficient angular momentum $L$.  Based on the sizes of the galaxies, there's some critical angular momentum below which they will collide, if they are also bound.
