Context Navigation

← Previous Post
Next Post →

Primary star frame system .VS. COM frame system

EVERYTHING IS ABOUT "AS IF"

If we locate the origin at the primary star. The Newton's law is:

F=\frac{G m M}{r^2}=m_{reduce} \omega^2 r

\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho \mathbf{v})=0

\frac{\partial\rho \mathbf{v}}{\partial t}+\nabla\cdot(\rho \mathbf{v} \mathbf{v})=\mathbf{f_1}+\mathbf{f_2}

\mathbf{f_2} is using m instead of m_{reduce}

r=r_1+r_2

r_1 is the distance from primary to COM

r_2 is the distance from secondary to COM

If we locate the origin at COM.

F=\frac{G m M}{r^2}=m \omega^2 r_2=m_{reduce} \omega^2 r

\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho \mathbf{v'})=0

\frac{\partial\rho \mathbf{v'}}{\partial t}+\nabla\cdot(\rho \mathbf{v'} \mathbf{v'})=\mathbf{f_1'}+\mathbf{f_2'}

So the newton's law does not really change if we just use F=\frac{G m M}{r^2}

for the fluid mechanics part, it explicitly has the following relation:

\mathbf{v}=\mathbf{v'}+\mathbf{v}_{com}

and \mathbf{v}_{com} is known - doing circular motion

Therefore, it is the mapping from COM frame to primary frame that is complicated. As long as we always work in the same frame, we are fine.

\frac{d\mathbf{r}}{dt}=\frac{d\mathbf{r'}}{dt}+\frac{d\mathbf{r}_{com}}{dt}

NOTE: All quantities with com subscript are known.

The trick here is to use m_{reduce} in kinematic equation but use m in fluid equations and newton's law. Thing will get more complicated if secondary accrete gas - need to update reduced mass every time step.

Actually you can view every parcel of gas as a body, and it will has its own reduced mass with respect to either star which is approximately its own mass. That justify why we don't need to change the density. If we have to change the density, serious things will happen - EOS changed and that will affect the dynamical picture.

astrobear

Context Navigation

Primary star frame system .VS. COM frame system

Comments

Download in other formats: