Can the Sun lose enough mass that Saturn's current velocity becomes escape velocity?

Question

Imagine that, through some cosmic phenomena not yet understood, the Sun 'burps' and ejects a vast amount of its mass into the cosmic void. A huge coronal discharge, perhaps. A pressure bubble inside that bursts. A mega internal explosion. The loss of mass is sudden and dramatic, but in a trajectory that does not traverse the planetary plane, and thus does not destroy any planets. Perhaps it occurs along the axis of the planetary plane. That is, the Sun does not 'burn out', go nova, or any other destructive end-of-life process, it just loses a substantial amount of mass. No other destructive radiation or other event that would immediately destroy the planets.

This loss of mass would result in a dramatic decrease in the gravitational pull of the Sun. This would affect all of the planetary orbits, as their escape velocity from the Solar System would decrease. If they kept their current velocity, I presume they would move further from the Sun.

A. How much mass would the Sun have to lose, in order for Saturn's current velocity to become its escape velocity from the Solar System? This is a tricky calculation and equation, as it has to account for the diminishing gravity of the Sun, not an increased velocity of Saturn. That is, it does not ask for the new velocity of Saturn sufficient to reach the escape velocity of Saturn from the existing Sun, but asks for the the maximum reduced mass of the Sun such that the current velocity of Saturn becomes its escape velocity.

The following are ancillary, but not essential, questions that might arise from answering A.

B. Is there any absolute principle of physics that would make this absolutely impossible?

C. Is it feasible that Saturn, along with its moons, could become an intragalaxy or even intergalaxy wanderer using this technique? The ultimate goal is to put a sentient self-sustaining colony on one or more of its moons, and then have it wander the Universe. How to give it the ability to sustain life on a moon for millions of years is another question not within the scope of this question.

D. Does it make more sense from the escape velocity perspective for my ultimate objective to consider another planet, such as Neptune or Jupiter? I need a planet with sufficient composition for it to become a source of power for the moons. Jupiter, for instance, naturally emits a very high level of radiation that could provide a source of energy for its moons as a substitute for the Sun, but again this is beyond the scope of this question.

What happens to the Sun because of the loss of this mass AFTER Saturn becomes a wanderer is not within the scope of this question.

How the Sun actually loses the mass is beyond the scope of this question. That it can somehow lose this mass is to be taken as a given assumption.

Answer 1

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.

Answer 2

Interesting. Not that I want to answer my own question, but I would like to sum up the math from both HDE 226868 and kingledion (to whom I acknowledge and thank very much for their input).

As a general rule of thumb, it looks like the mass of an orbited body has to be reduced by about 50%, or half, in order for an existing orbital velocity of a satellite to become the escape velocity of the satellite to the reduced orbited mass.

EDIT

That is, when the Sun's mass decreases by 50%, EVERY planet's orbital velocity will become its escape velocity, at pretty much the same time. Strange, but apparently true.

End Edit

This could be a useful rule of thumb in so many different situations.

How it gets reduced by 50%, of course, is another question.

Answer 3

Even though @AlexP is right about the mass still existing, it's believable enough that -- unless you're writing Hard SF -- you can just assert that the Sun -- for example -- burped out 20% of it's mass and off zoomed Saturn.

For more realism, have the mass get burped in the opposite direction of where Saturn is. Saturn (and all the other planets...) gets pulled towards the new epicenter and then flies out into interstellar space.

Answer 4

The math already presented by HDE 226868 and kingledion is bang on so I won't repeat it, but I feel they have neglected one possibility. The mass of a star can theoretically be reduced artificially through the process of star lifting so it may be possible for an arbitrarily advanced civilisation to pull enough material out of the sun, and displace it from the solar system, that Saturn would exceed escape velocity.