A naive first calculation

The formulas for orbital velocity and escape velocity are $$v_o=\sqrt{\frac{GM}{r}},\quad v_e=\sqrt{\frac{2GM}{r}}$$ I get $v_o=9.6\text{ km/s}$ for Saturn. For this to equal $v_e$, the Sun's new mass would have to be $0.5M_{\odot}$, if we neglect the mass of the ejected gas. By doing the algebra, you can see that this number is the same for any mass and any orbital radius. In short, if Saturn achieves escape velocity because of solar mass loss, so should the other planets (assuming circular orbits).

A more realistic model

Here's why it's actually important that we establish how the Sun loses mass. As others have said, the ejected matter will also cause a gravitational potential, which will increase escape velocity, and thus influence Saturn's orbit.

I'm imagining that in a dramatic but compressed asymptotic giant branch phase, the Sun is losing mass at a rate of $\sim10^{-4}M_{\odot}\text{ yr}^{-1}$; its wind is isotropic, sending material streaming away from the star in all directions. I'll ignore the fact that strong winds from evolved Sun-like stars can ablate planets. We can model the density of the wind by $$\rho_{gas}(r)=\frac{\dot{M}}{4\pi r^2v(r)}$$ where $\dot{M}$ is the mass loss rate and $$v(r)=v_{\infty}\left(1-\frac{R_*}{r}\right)^\beta$$ with $R_*$ being the radius of the star. For massive stars, we normally assume that $\beta\approx1$. We can then find the gravitational potential by solving Poisson's equation: $$\nabla^2\Phi_{gas}=4\pi G\rho_{gas}$$ From this, we can determine the escape velocity: $$v_e(r)=\sqrt{2|\Phi(r)|}$$ where $\Phi(r)=\Phi_{gas}+\Phi_{\odot}$, and $\Phi_{\odot}$ changes in time as the Sun loses mass. You should be able to work backwards from here to determine the mass-loss rate and wind terminal velocity, given a desired escape velocity (Saturn's current orbital velocity, $9.6\text{ km/s}$).

Calculating the potential for other mass-loss scenarios is beyond me, because I don't know the proper density distribution; I can only talk about modeling stellar winds. I suspect that bipolar jets would contribute very little, as they are away from the orbital plane and would presumably be moving very quickly (even compared to $v_{\infty}$, which can be $\sim2000\text{ km/s}$ for massive stars, though perhaps a mere $\sim300\text{ km/s}$ for a Sun-like star). Similarly, a massive coronal mass ejection or super-super-superflare wold be hard for me to model.

Doing the math

Let's make some (justified) assumptions:

• At Saturn's orbit, $R_*/r\ll 1$, so we can use the binomial approximation to write $$\rho_{gas}(r)\approx\frac{\dot{M}}{4\pi v_{\infty}}\left(r^{-2}+\frac{R_*}{r^3}\right)$$
• The wind is isotropic, so $$\nabla^2\Phi_{gas}=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\Phi_{gas}}{dr}\right)=4\pi G\rho_{gas}(r)$$

This gives us a potential of the form $$\Phi_{gas}(r)=\Phi_0-\frac{G\dot{M}}{v_{\infty}}\left(\frac{R_*}{r}+\frac{R_*\log (r/r_e)}{r}-\log(r/r_e)\right)-\frac{D}{r}$$ for some constants $\Phi_0$, $r_e$, and $D$. Therefore, $$v_e(r,t)=\sqrt{2\left|\frac{G(M_{\odot}-\dot{M}t)}{r}+\Phi_{gas}(r)\right|}$$ at time $t$. Choosing the correct $\dot{M}$, $v_{\infty}$ and constants will get you the precise situation you want.

posted over 6 years ago

HDE 226868‭

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