Solar recycling or: How to keep your star from dying

A star's life ends when it can no longer undergo fusion at its core. For massive stars, this often happens when the core is made largely of iron, which can be fused (and is) in small amounts, but only endothermically. The products of nuclear fusion are, at this point, like carbon monoxide: it's not that their presence is toxic to the star, but it prevents the star from getting the fuel it needs - sort of like how CO molecules binding to hemoglobin make it difficult for a human to get the oxygen it needs. They take up space. Therefore, if you could remove those products, hydrogen could take their place at the core, and the death of the star would be offset.

I very much agree with Ender Look on one point: convection is key. It allows you to transport heavy elements from deep within the star to its surface, while mixing hydrogen back into the core. This happens in the outer regions of all stars, but in low-mass stars(say, less than $\sim0.3M_{\odot}$), this convective region reaches all the way to the core. The Sun's convective zone ends in a place called the tachocline, which occurs at about $0.7R_{\odot}$. Below this point, the star is stable against convection. In particular, something called the Schwarzschild criterion is satisfied: $$\frac{3}{64\pi\sigma G}\frac{\kappa LP}{MT^4}<1-\frac{1}{\gamma_{\text{ad}}}$$ where $\kappa$ is the opacity and $\gamma_{\text{ad}}$ is the adiabatic index. We can assume that $\gamma_{\text{ad}}=5/3$. I decided to try to model the tachocline on my own by determining where the following is satisfied: $$\frac{3}{64\pi\sigma G}\frac{\kappa LP}{MT^4}-\left(1-\frac{1}{\gamma_{\text{ad}}}\right)<0$$ I used numerical simulations by John Bahcall (in particular, the model designated (BS2005-AGS,OP)). I calculated the opacity via Kramer's opacity law, including opacity contributions from free-free and bound-free absorption, as well as electron scattering: $$\kappa_{\text{ff}}=3.68\times10^{22}g_{\text{ff}}(1-Z)(1+X)\frac{\rho}{\text{g cm}^{-3}}\left(\frac{T}{K}\right)^{-7/2}\text{ cm}^2\text{ g}^{-1}$$ $$\kappa_{\text{bf}}=4.634\times10^{25}\frac{g_{\text{bf}}}{t}Z(1+X)\frac{\rho}{\text{g cm}^{-3}}\left(\frac{T}{K}\right)^{-7/2}\text{ cm}^2\text{ g}^{-1}$$ $$\kappa_{\text{es}}=0.2(1+X)\text{ cm}^2\text{ g}^{-1}$$ Assuming that $g_{\text{ff}}\approx g_{\text{bf}}\approx1$ and $t\approx10$, I get the following plot of opacity contributions:

I then plotted the difference of the two sides of the inequality:

This places the tachocline at about $0.82R_{\odot}$ - an overestimate, but not by much.

Notice that for the two major components of the opacity, $\kappa\propto\rho T^{-7/2}$. Therefore, if you increase the density or decrease the temperature, $\kappa$ will increase and the convective region will move inwards. This is what happens during something called a dredge-up, affecting stars that are off the main sequence. Say we cut the temperature to $0.75$ times its current value at all points in the outer half of the Sun. Then we find that the convective envelope can extend to about $0.72R_{\odot}$ - progress. Now, the core extends only to $0.25R_{\odot}$, so we would need to dramatically increase the opacity much deeper in the star to further lower the tachocline.

Once we can figure this out, all that remains is to determine how to increase the efficiency $\lambda$ of this artificial dredge-up - that is, how much of the newly-produced heavy elements are cycled to the surface. Ideally, we'd have $\lambda\approx1$, and typical dredge-up efficiencies vary.

The solution I've come up with is almost counterintuitive: Add more heavy elements. Notice that $\kappa_{\text{bf}}$ - the major source of opacity - is proportional to $Z$. This means that by increasing the amount of heavy elements in the outer layers of the star, we can extend the convective zone down towards the core. Given that for the Sun, $Z\simeq0.001$, changing this shouldn't change $X$ and $Y$ by a significant amount. In fact, $Z$ is so small that there's only about one Jupiter mass of heavy elements in the Sun. I decided to adjust $Z$ by various amounts, and looked at how deep the convective zone extended.

It seems that by adding about Jupiter masses worth of heavy elements, we can extend the tachocline to at least $0.6R_{\odot}$, if not deeper. Remember, of course, that my original calculations underestimated tachocline depth, meaning that in reality, this could be enough to bring the convective zone to the edge of the core.

You might notice that around $0.5R_{\odot}$, it seems that my computations produces a convective region extending to the core even at low metallicities. I believe this zone doesn't exist in those low-metallicity models. So why does it show up? Well, notice how $X$ and $Y$ change deep in the Sun:

Inside about $0.2R_{\odot}$ - and, to an extent, inside $0.5R_{\odot}$ - there is non-negligible gradient of both $X$ and $Y$, which translates to a non-negligible gradient of the mean molecular mass, which means our stability criterion should actually be something called the Ledoux criterion, which takes concentration gradients into account. If I accounted for this, I believe this supposed convective region would disappear - except in the high-metallicity models.

My guess is that adding roughly 30 Jupiter masses (likely less, given that we don't need to increase the metallicity in the core - just the region directly outside it) worth of heavy elements would increase the opacity to the extent that the tachocline would reach the core, allowing for mixing and eventually heavy element transport to the surface. The details of how you get all of these heavy elements back out further compounds the problem, but I believe that if you could trigger a burst of fusion, that could lead to an artificial dredge-up, as I discussed before.

Rinse and repeat, periodically - perhaps every few billion years or so, just to be safe. Honestly, 30 Jupiter masses every few billion years isn't too much to ask of a Type II civilization, is it (depending on what resources are in the planetary system, of course)?

posted over 5 years ago

HDE 226868‭

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