# Effects of a larger value for the Gravitational Constant?

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?

• Well, if you had an individual star that was many times smaller with 100x gravity, it should work the same as a normal star in our universe, as far as I can imagine. But globally there would be a lot more black holes and those little stars wouldn't have a chance to form. Perhaps everything would be black holes. Also there would have been some problems with an expanding universe with 100x gravity, unless you compensate for that. Jul 26 at 8:49
• It is a very interesting question, and I for one would be interested in reading high-quality answers from people who know their physics. Jul 26 at 10:42
• @JBH Tag changed. Jul 26 at 18:09
• Be sure to read Raft by Stephen Baxter... Jul 26 at 18:20
• I agree with @AlexP; the value of $G$ has nothing to do with the speed of gravity, which general relativity predicts (and LIGO observations confirm, with standard uncertainties) is the same as the speed of light. I'm not sure there's much of an issue, and I also don't really see an issue with using the hard science tag. Jul 26 at 19:07

## 2 Answers

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.

• Three side questions: (a) Would this change the habitability zone location for a given star type? (b) Does this shift down in stellar mass mean a similar shift down in average planetary mass? (c) Would this make gas giants much more rare in such a universe? or would it result in a greater number of multiple-star systems? Jul 26 at 21:43
• @JBH (a) The habitable zone depends on luminosity, so yes, for a given stellar mass. (b) Yes, I believe so. (c) Yes, almost certainly we'd see fewer giant planets - we've already seen a lack of gas giants around red dwarfs. Jul 26 at 21:56
• If a universe with G increased by two orders of magnitude would have stars of lower masses, would a universe with a, would a universe with G decreased by two orders of magnitude have stars of higher masses? Jul 27 at 9:06
• @AndersGustafson That's a good question, and while I'm pretty certain it's a yes, I suspect it's a little more complicated. Using the above logic, I'd expect masses of $\sim100\mathrm{-}10^5M_{\odot}$, but at the upper end I do wonder whether molecular clouds would even be capable of reaching those masses, particularly with gravity being weaker, so the upper limit might be truncated. There's also the question of whether stellar fusion would even be the dominant method of energy generation in such a universe. Jul 27 at 18:07
• I'm attempting the equations in your answer, and I was wondering if you could check my math? For a test case of $M = 0.00008 M_⊙$, I'm getting a $D_{rad}$ value of $-6.5$ and an $α_R$ of $-0.3077$. Raising M to this $α_R$ gives me a radius of $18.2216 R_⊙$, which doesn't seem right. Jul 27 at 21:46

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.