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Why Gravity is So Weak

By Martin Rees, Ph.D.   Fri, Jun 17, 2011


Stars can be defined as gravitationally-bound fusion reactors. A very simple argument allows us to work out how big they are -- how many atoms you'd have to pack together to make one.

Gravity is a very, very weak force. In a hydrogen molecule it pulls the two protons together with a force about 36 powers of ten weaker than the electric force between them. But in any large object, positive and negative electric charges almost cancel out. In contrast, everything has the same sign of gravitational charge, so gravity inexorably gains importance in large objects. What does it take for gravity to win?



The answer involves some arithmetic-- but nothing complicated. Suppose you assemble progressively larger lumps containing 10, 100, 1000 atoms, and so on. The 24th would be the size of a sugar lump; the 40th would be the size of a mountain or a small asteroid.

The effect of gravity on each atom -- how strongly gravity binds it to all the others in the lump -- goes up in proportion to the total number of atoms but down by their average distance from each other. For each 1000-fold increase in mass, the importance of gravity goes up 100 fold.

(This is because, though the number of atoms goes up by 1000, their average distance from each other goes up by 10). Despite its initial handicap, amounting to 36 powers of 10, gravitational forces becomes dominant when more than about 10 to the power 54 protons are packed together (36 being two thirds of 54) -- that's a number that can be written as one followed by 54 zeros. This mass is about the same as that of Jupiter, the biggest planet in our Solar System. To become a star, a body must be about a hundred times more massive still -- so that it can hold itself together, gravitationally, even when its center is hot enough for nuclear fusion to occur.

Gravity eventually wins -- but because it is so weak it only triumphs on very large scales. The Princeton physicist Robert Dicke was the first to emphasize the key point that stars are so big because gravity is so weak -- because the ratio of the electrical and gravitational forces within atoms is such a huge number. . Dicke also estimated the time it takes for heat to diffuse out of a star, showing that this time is long, implying that stars have long lives as well as being big, because this same ratio is so large.

It's amusing to ask what the universe would be like if gravity weren't quite so weak. Suppose, for example, that gravity was 'only' 26 rather than 36 powers of ten weaker than electric forces in atoms- but the properties of the atoms themselves were unchanged. Atoms and molecules would behave just as in our actual universe, but objects would not need to be so large before gravity became competitive with the other forces. In this imagined universe stars would contain a million billion times fewer particles than the sun does. If these stars had planets around them, they would be smaller than the actual planets in our Solar System by the same factor, but gravity on their surfaces would pull far more strongly than on Earth. Strong gravity would crush anything larger than an insect on hypothetical miniplanets around these miniSuns. But more severe still is the limited time. Instead of living for 10 billion years, a miniSun would last for about 1 year. and would have exhausted its energy before even the first steps in organic evolution had got under way. The actual scaling isn't quite this simple: mini-stars in this strong-gravity universe would, for instance, have somewhat hotter surfaces than actual stars. But the outlook for complex evolution would plainly be less propitious, because there is less space and less time for its operation. Any creatures bigger than insects living on a planet with an atmosphere, would be crushed by gravity.

There would be fewer powers of 10 between the lifetime of stars and the basic microphysical timescales for physical or chemical reactions so no such creatures could have had time to emerge via evolution anyway.

If gravity weren't so weak, the universe couldn't contain such a multilayered hierarchy of structures, and wouldn't allow time for complex evolution. So gravity is crucial in the cosmos -- but the weaker it is (provided it isn't zero) the grander and more complex can be its manifestations.

By Martin Rees, Ph.D.

Martin Rees, Ph.D.

Master of Trinity College and Astronomer Royal. Former Director of the Institute of Astronomy. My most recent books are 'From here to Infinity' (Profile books) and 'Gravity' Fatal attraction' (CU Press)