Rocket Equation from Newton’s 3rd Law, Force of Gravitation, and dp/dt

Question

I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth.

  $$ F - F_{gravity} = \dfrac{\mathrm{d}p}{\mathrm{d}t} \\ F - \dfrac{G \, m_e \, m_r}{r^2} = v \, \dfrac{\mathrm{d}m}{\mathrm{d}t} + ma\\ $$  

$F$ = Force created by fuel (at time t)

$G$ = Gravitational Constant

$m_e$ = Mass of Earth

$m_r$ = Mass of rocket (at time t)

$r$ = Distance between Earth and rocket (at time $t$)

$v$ = Velocity of rocket relative to Earth (at time $t$)

$dm/dt$ = Instantaneous rate of change of mass of rocket (at time $t$)

$m$ = Also mass of rocket (at time $t$)

$a$ = Instantaneous acceleration of rocket (at time $t$, equal to $dv/dt$)

 

Is my equation correct for a standard rocket? Would dm/dt be negative as the rocket is losing mass over time?