Changes between Version 11 and Version 12 of u/erica/RadFeedback

Timestamp:

11/19/15 13:44:48 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/RadFeedback

@@ -7 +7 @@
 == Accretion Luminosity == 
 
-The amount of energy deposited into the kernel around a sink is intuitively given by the accretion energy. As infalling material hits the surface of the star, its kinetic energy is converted to heat. For spherical symmetry, a gas parcel freely falling to the star from infinity will have its kinetic energy and gravitational energy balanced once the parcel reaches the star:
+The amount of energy deposited into the kernel around a sink is intuitively given by the accretion energy. As infalling material hits the surface of the star, its kinetic energy is converted to heat. For spherical symmetry, a gas parcel starting from rest and freely falling to the star from infinity will have its kinetic energy and gravitational energy balanced once the parcel reaches the star:
 
-[[latex($\frac{1}{2} m v_{ff}^2 = \frac{GmM}{r}$)]]
+[[latex($\frac{1}{2} m v_{ff}^2 = \frac{GmM_{*}}{R_{*}}$)]]
 
 As the material strikes the surface of the star (i.e. is accreted) the kinetic energy is converted to heat. For an accretion rate [[latex($\dot{m}$)]], the rate at which this heat is produced, or the luminosity L, is given by:
 
-[[latex($ L = \frac{1}{2}\dot{m} v_{ff}^2 = \dot{m}\frac{GM}{r}$)]]
+[[latex($ L = \frac{1}{2}\dot{m} v_{ff}^2 = \frac{G\dot{m}M_{*}}{R_{*}}$)]]
 
-To first order, 
+However, in our simulations the gas parcel isn't falling into the sink from infinity, but rather from some distance r away from the sink. At this radius it has some kinetic and gravitational potential energy, as it does once it falls to the surface of the star. Energy conservation gives:
+
+[[latex($\frac{1}{2}m v(r)^2 - \frac{GmM_{enc}}{r} = \frac{1}{2}m v(R_{*})^2 - \frac{GmM_{*}}{R_{*}}$)]]