Effects of a larger value for the Gravitational Constant?

Question

Assuming otherwise-identical physics, what would be the effects on stars and planets if the value of the Gravitational Constant was, for example, 100 times greater than in reality?

In particular, how it affect the mass/luminosity/radius/temperature relationships of main-sequence stars? Would stars as we know them even be possible?

Answer 1

There's going to be a good deal of educated guesswork in the latter part of this answer, so if you're able to find some big holes in it, please do, and let me know.

First, let's address one fundamental question: Can stars even exist if we increase $G$ by two orders of magnitude? The answer seems to be yes; Adams 2008 (a great paper if you want to take a look) considers this question and indicates that $100G$ still lies in the region of parameter space where the p-p chain remains possible even without modifications to other constants like the fine structure constant $\alpha$ or the strengths of the nuclear forces (see Figure 5).

One significant change we'd see is that the mass ranges of stars would be shifted downwards. The above paper shows that the minimum and maximum masses of a star are both proportional to $G^{-3/2}$, so the lower limit would now be $\sim0.00008M_{\odot}$ (only 27 Earth masses!) and the upper limit would be somewhere around $0.1\mathrm{-}0.3M_{\odot}$. Below this range, hydrogen fusion would be impossible; above it, radiation pressure would blow a star apart. Fortunately, the characteristic mass at which an interstellar gas cloud would fragment (the Jeans mass) is also proportional to $G^{-3/2}$, so clouds will still form objects in the right range to form stars. The major difference to stellar populations is that, with the average mass of a star now lower by three orders of magnitude, the number of stars in the universe should increase by roughly three orders of magnitude.

Regarding the various relationships between mass and other quantities (radius, luminosity, etc.): The exponents in these relationships depend on the opacity, equation of state and energy generation mechanism within the star. For instance, if we guess that $$R\propto M^{\alpha_R},\quad\rho\propto M^{\alpha_{\rho}},\quad L\propto M^{\alpha_L},\quad T\propto M^{\alpha_T}$$ it turns out we can use the equations of stellar structure and some other assumptions to derive values for all four exponents.$^{\dagger}$ There are six relevant parameters which can be used to calculate the exponents:

• $\chi_T$ and $\chi_{\rho}$, which describe how pressure relates to density and temperature through $$P\propto\rho^{\chi_{\rho}}T^{\chi_T}$$ If we assume the star is an ideal gas, $\chi_T=\chi_{\rho}=1$.
• $n$ and $s$, which describe how the opacity depends on density and temperature through $$\kappa\propto\rho^nT^{-s}$$ If electron scattering is the dominant source of opacity, $n=s=0$. If Kramers' opacity dominates (at low temperatures), $n=1$ and $s=3.5$.
• $\nu$ and $\lambda$, which can be determined by the energy generation mechanism at work. For example, the p-p chain requires that $\nu=4$ and $\lambda=1$, while the CNO cycle requires that $\nu=15$ and $\lambda=1$ and the triple-$\alpha$ process requires that $\nu=40$ and $\lambda=2$.

In our case, dealing with low-mass stars, the ideal gas law should still hold, so we still have $\chi_T=\chi_{\rho}=1$. What about the other factors? Will we still have the p-p chain? Well, we'd need to stay below core temperatures of $\sim$18 million Kelvin. An estimate using either the virial theorem or a simply analytical density model tells us that the central temperature and central pressure scale like $$T\propto\frac{GM}{R},\quad P\propto\frac{GM^2}{R^4}$$ For one of our most massive stars, $M=0.1M_{\odot}$. Most stars have $\alpha_R$ a bit less than 1, so plugging in some numbers, it does seem like a star around this mass with $G$ increased by two orders of magnitude could indeed utilize the CNO cycle. We will, then, have many of our "high-mass" stars using the CNO cycle even though they'd either use the p-p chain in our world or be incapable of nuclear fusion. The truly low-mass stars would stick to the p-p chain, I assume.

Finally, there's the question of opacity. More massive stars should be dominated by electron scattering, while less massive stars should be dominated by Kramer's opacity. We'd see the same division we see in our universe. On the other hand, convection sets in at low masses and complicates some of our above assumptions, and I don't truly know how that would affect things.

In short, the various relationships likely wouldn't look too different from the way they look in our universe; we'd just see them applied to drastically different mass ranges. The precise values would also exhibit additional scaling because there will be an explicit multiplicative factor of $G$ in some of them.

---

$^{\dagger}$Stellar Interiors, Hansen, Kawaler and Trimble.

Answer 2

Gravitation is weak. Very weak. Even if boosted by the factor of 100, it would not play any role in the early stages of the universe.

On the other hand, during the Dark Ages it dominates, and the value of $\gamma$ becomes important, specifically in the star formation process. The universe is pretty much a free hydrogen at $4^{\circ} K$, and the attraction radius is defined by the equilibrium $\dfrac{\gamma}{R} \sim kT$.

Gravity 100 times stronger increases the radius 100 times, so the typical star would accrete $10^6$ times more material. I am not versed enough in the stellar evolution theory to predict the consequences. I don't know how the main sequence would look like - and what the main sequence would be. It looks very likely that white dwarfs would be sorely missing.