  A tidal disruption event (TDE) -- when a star is destroyed by the immense
gravitational field of a supermassive black hole -- transforms a star into a
stream of tidally-shredded debris. The properties of this debris ultimately
determine the observable signatures of TDEs. Here we derive a simple,
self-similar solution for the velocity profile of the debris streams produced
from TDEs, and show that this solution agrees extremely well with numerical
results. Using this self-similar solution, we calculate an analytic,
approximate expression for the radial density profile of the stream. We show
that there is a critical adiabatic index that varies as a function of position
along the stream above (below) which the stream is unstable (stable) to
gravitational fragmentation. We also calculate the impact of heating and
cooling on this stability criterion.
