## Gravity of a Five-Body System

Let's look at various examples of gravity that we see in the solar system. We have the Moon orbiting the Earth, and our sphere orbits the Sun (along with the other planets). While the system is always changing, it is, for the most part, a stable one. But (in an orbital system of two similarly massed objects), if a third object of comparable mass enters that system, to put it lightly, it creates chaos. Because of competing gravitational forces, one of the three objects will be ejected and the remaining two will be in a closer orbit than before. Nevertheless, it will be more stable. All of this results from Newton’s Theory of Gravity, which as an equation is F = m1m2G/r^2, or that the force of gravity between two objects equals the gravitational constant times mass of first object times mass of second object divided by distance between objects squared.

It is also a result of the Conservation of Angular Momentum, which simply states that the total angular momentum of a system of bodies must remain conserved (nothing added nor created). Because the new object enters the system, its force on the other two objects will increase the closer it gets (for if distance decreases, then the denominator of the equation decreases, increasing the force). But each object pulls on the other, until one of them has to be forced out to return to a two system orbit. Through this process, angular momentum, or the tendency of the system to continue as is, must be conserved. Since the departing object takes some momentum away, the remaining two objects get closer. Again, that decreases the denominator, increasing the force the two objects feel, hence the higher stability. This entire scenario is known as a “slingshot process” (Barrow 1).

But, what about two two-body systems in near proximity? What would happen if a fifth object entered that system? In 1992, Jeff Xia investigated and discovered a counter-intuitive result of Newton’s gravity. As the diagram indicates, four objects of the same mass are in two separate orbiting systems. Each pair orbits in the opposite direction of the other and are parallel to each other, one above the other. Looking at the net rotation of the system, it would be zero. Now, if a fifth object of a lighter mass were to enter the system in-between the two systems so that it would be perpendicular to their rotation, one system would push it up into the other. Then, that new system would also push it away, back to the first system. That fifth object would go back and forth, oscillating. This will cause the two systems to move away from each other, because the angular momentum has to be conserved. That firth object receives more and more angular momentum as this motion goes on, so the two systems will move further and further away from each other. Thus, this overall group “will expand to infinite size in finite time!” (1)

## Doppler Shifting Time

Most of us think of gravity as the result of mass moving through spacetime, generating ripples in its "fabric." But one can also think of gravity as a redshift or a blueshift, much like the Doppler effect, but for time! To demonstrate this idea, in 1959 Robert Pound and Glen Rebka performed an experiment. They took Fe-57, a well established isotope of iron with 26 protons and 31 neutrons that emits and absorbs photons at a precise frequency (roughly 3 billion Hertz!). They dropped the isotope down a 22 meter fall and measured the frequency as it fell towards the Earth. Sure enough, the frequency at the top was less than the frequency of the bottom, a gravitational blueshift. This is because gravity compacted the waves that were being emitted and because c is wavelength times frequency, if one goes down the other goes up (Gubser, Baggett).

## Eclipses and Space-Time

In May 1905, Einstein published his special theory of relativity. This work demonstrated, amongst other work, that if an object has sufficient gravity then it can have an observable bending of space-time or the fabric of the universe. Einstein knew that it would be a hard test, because gravity is the weakest force when it comes to small-scale. It would not be until May 29th, 1919 that someone came up with that observable evidence to prove Einstein was right. Their tool of proof? A solar eclipse (Berman 30).

During an eclipse, the Sun’s light is blocked out by the Moon. Any light that comes from a star behind the Sun will have its path bent during its pass near the Sun, and with the Moon blocking out the Sun’s light, the ability to see the starlight would be easier. The first attempt came in 1912 when a team went to Brazil, but rain made the event unviewable. It ended up being a blessing because Einstein made some incorrect calculations and the Brazilian team would have looked in the wrong place. In 1914, a Russian team was going to try for it but the outbreak of World War I put any such plans on hold. Finally, in 1919 two expeditions are underway. One goes to Brazil again while the other goes to an island off the coast of West Africa. They both got positive results, but barely. The overall deflection of the starlight was “about the width of a quarter viewed from two miles away (30).

An even harder test for special relativity is not only the bending of space but also time. It can be slowed down to an appreciable level if enough gravity exists. In 1971, two atomic clocks were flown up to two different altitudes. The clock closer to the Earth did end up running slower than the clock at the higher altitude (30).

Let’s face it: we need gravity to exist, but it has some of the strangest influences we have ever encountered in our lives and in the most unexpected ways.

## Works Cited

Baggett, Jim. Mass. Oxford University Press, 2017. Print. 104-5.

Barrow, John D. 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. New York: W.W. Norton &, 2009. Print.

Berman, Bob. “A Twisted Anniversary.” Discover May 2005: 30. Print.

Gubser, Steven S and Frans Pretorius. The Little Book of Black Holes. Princeton University Press, New Jersey. 2017. Print. 25-6.

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