Speculative physics: G doubled

Problem

Speculative physics: $G$ doubled. What would change in the Solar System and on Earth if the universal gravitational constant $G$ doubled tomorrow? Discuss the effects on orbits, periods, weight and stars.

Solution

The gravitational force is $F = G\,\dfrac{Mm}{r^2}$: if $G \to 2G$ all attractions double at a fixed distance. Let’s analyse each case.

Orbits. For a circular orbit the gravitational force provides the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, i.e. $v^2 = \dfrac{GM}{r}$. Planets currently move at fixed speeds; with $G$ doubled the force towards the Sun exceeds what is needed to keep them on their current orbit, so they would fall inward: the orbits would shrink.

Periods. From Kepler’s third law $T = 2\pi\sqrt{\dfrac{a^3}{GM}}$: at fixed semi-major axis $a$, if $G \to 2G$ then $T \to \dfrac{T}{\sqrt 2}$. Periods would shorten by a factor of $\sqrt 2$ (shorter years).

Weight. At the surface $g = \dfrac{GM_T}{R_T^2}$: it would double. $g' = 2g \approx 2 \cdot 9{,}8 \approx 20\ \text{m/s}^2$ Every body’s weight would double: enormous difficulty walking, supporting structures, muscles and bones inadequate.

Stars. A star is in hydrostatic equilibrium between internal pressure and self-gravity. With a larger $G$, gravitational compression is more intense: hotter cores, faster fusion, shorter stellar lifetimes. The collapse thresholds (the Chandrasekhar limit $M_\text{Ch} \propto G^{-3/2}$) drop: white dwarfs and neutron stars would form at lower masses. The cosmos would be radically different.

$\ev{\text{narrower orbits};\quad T \to T/\sqrt2;\quad g \approx 20\ \text{m/s}^2;\quad \text{short-lived stars}}$

Links

Topics: Gravitation Concepts: Law of universal gravitation · Kepler’s laws · Weight force Skills: Analysis of limiting cases and speculative physics