Still a bit low. We want: \frac{G m_1 m_2}{r^2} = \frac{k_e q_1 q_2}{r^2}
Cancel the r's and set q_1=q_2: q = \sqrt{\frac{G}{k_e} m_1 m_2}
That’s q = 5.71 \times 10^{13} C. Since 1 \ C = 6.24 \times 10^{18} e and protons are 1.6726 \times 10^{-27} kg/e, we get 1 \ C \ \text{protons} = 1.044 \times 10^{-8} kg.
Multiplied all out, we get 596,000 kg of protons per body. Which is still pretty tiny compared to the 5.972 \times 10^{24} \ kg mass of the Earth. It’s only 325 kg if you switch to electrons.
A good way to demonstrate the weirdness of gravity is to take a sphere of some material of the same density as the Earth, say about the size of a basketball, and then another sphere proportionately the same size as the moon, and set it down on a very, very big table proportionately as far away as the moon is from Earth (which is actually surprisingly far in relation to their sizes). And then see what happens.
What? Nothing happens at all? Why are the two not attracted to each other by gravity? Why do they not collide on the table top if the moon is not in a stable orbit?
I think a lot of the apparent mystery of gravity has to do with the fact that’s it’s an incredibly weak force acting in an environment in which there are no other forces, so as weak as it is, it’s the force that determines how objects in space behave, because there are no other forces to compete with it.
There must be some framework that governs how both gravity and quantum mechanics function, and which can apply in contexts where both gravity and quantum mechanics are relevant. Whether it is possible for humans to ever accurately describe that framework, however, is an open question.
Gravity isn’t simply weak. In a sense it’s the only ultimately strong force (and here I register my complaint at the use of “strong force” to name a fundamental force). Look at black holes – gravity is so strong there that the bottom drops out of geometry itself. None of the other three forces do anything like that, do they?
If it helps, the fundamental force is actually the color force (which is still a silly name, but at least less likely to be misinterpreted). The “strong force” is actually just a residual manifestation of it, like the van der Waals force is to electromagnetism.
Though for that matter, I also prefer the term “interaction” to “force”, in this context. “Force” makes sense when you have basically the same objects coming out of an interaction as went in, with the major result of the interaction being changes in their momentum. But in most of the interesting “strong” and “weak” interactions, the major result of the interaction is that the identities of the particles are changed entirely.
It would certainly make getting to space far more difficult. I did an exploration of this years ago for a similar question and it gets pretty onerous quickly for chemical propulsion because you are limited by the specific energy available from the combustion of propellants and the additional mass of extra propellants which have to be lifted to achieve a higher orbital speed. Assuming the same density as Earth, the orbital and escape speed from “Super-Earth” scales at about a and exponent of 2/3 to the increase in gravity, such that at 1.5 x (1 g) the factor of increase in orbital speed is ~1.31, at 2.0 g it is ~1.59, and so forth. But, kinetic energy scales at a square to speed, so comparable energies are the same increase are ~1.72 and ~2.52. Applying the Tsiolkovsky rocket equation shows that the mass fraction of final over initial mass scales as an exponential, so a 1.5 g the mass of the rocket increases by almost a factor of 4, at 2.0 g it increases by a factor of 5, 3.0 g by a factor of 8, and so forth, assuming that you are comparing single stage to orbit (SSTO) launch vehicles.
Of course, we don’t have SSTOs, and would be just barely possible for a theoretical SSTO to put a relatively payload in orbit using a high performance chemical rocket; our launch vehicles to orbit are multistage specifically to take advantage of the rocket equation by expending unneeded inert mass of empty tanks, engines, and other structure, and also to optimize the performance of the rocket at altitude by configuration of nozzle geometry, propellant selection and mix ratio. The reality is that a rocket carrying the Apollo CSM to Low Earth Orbit would be something like eight times more massive at 1.5 g, almost ninety times more massive at 2.0 g, and hundreds of times larger (and would have to be broken into more stages) above that. At around 10 g, the required propellant alone to achieve orbit is some appreciable fraction of the mass of the planet which is obviously impossible.
This has been used by some to argue why the Earth is ‘uniquely perfect’ for an advanced space-faring species, and that it could not evolve on a much larger planet (and a smaller planet would not be able to hold onto sufficient atmosphere to create “Earth-like” conditions). But it also provides a counterargument that while Earth-like worlds capable of supporting a terrestrial civilization capable of space travel might be rare, civilizations that develop on the smaller mass moons of gas giants could be common, and with their much lower gravity and the ability to use other worlds for propellant-less maneuvers could facilitate space exploration and even access to and utilization of space resources. Of course, a civilization that developed in an ocean under an ice crust or in a frigid environment like Saturn’s moon Titan would have a quite different approach to technology and industrial development. But then, outside of science fiction, nobody expects that an advanced alien species will be humanoid and have a similar evolutionary and developmental path to industrial society as humanity.
Of course, the reality is probably even worse than that, because a larger “super-Earth” would probably be more dense. Even if it’s the same composition, it’d be compressed more. So the radius would be smaller than that of the same-density world, and so rockets would be starting off even deeper in the gravity well.
The way I try to express this in a more intuitive manner is that precisely because gravity is so strong, the planet we were born on is moving through space really fast (if it weren’t that fast, it wouldn’t be where it is but rather fall towards the Sun). So each time we leave our planet we’re still moving really fast relative to the Sun (because we still have the speed of the planet we’re coming from), and need to slow down to approach the Sun.
Assuming you built your model table perfectly perpendicular to the Earth’s gravitational potential at each point the table would be functionally flat and functionally horizontal. Despite having some detectable undulations geometrically speaking. Which is to say that given this appropriate table Earth’s gravity would absolutely, positively, and completely cancel out of the equation at all places and times, leaving you with an essentially two-dimensional system consisting only of mock-Earth and mock-Moon. Just like you alternate model of two sphere’s floating in idealized space free of other gravitational influences.
The actual problem with the tabletop mock model is the second order effects of Earth’s gravity, not the gravity itself.
You’re performing this experiment in an environment where Earth gravity causes rolling friction and “stiction” between the table and the mock-Earth and mock-Moon. The friction, and the angular momentum requirement it imposes on the system is what prevents them from rolling towards one another.
If we magick away the friction so they can slide freely on the frictionless table we encounter the second second-order problem caused by Earth gravity.
There’s an atmosphere in the way of our mocks and as soon as they start to move under their mutual gravitation, they’ll begin encountering aerodynamic drag that will limit their top speed to a nearly infinitesimal number. Compared to the case in free space where they’d accelerate exponentially towards one another until the Big Clunk as they collide on the experimental table.
Now once you magick away both friction and atmospheric drag, then your tabletop model would behave exactly as would your “two spheres in otherwise gravitation-less space” model.
Gravity: it’s just a pain in the ass overall. Always adding confounding factors to the mix.
True, and perhaps the weirdness can be restated in terms of the fact that in comparison to all the other fundamental forces, the apparent force of gravity is extremely weak relative to the mass of the objects that exert them.
But your point about the distortion of spacetime geometry is well taken, and provides the answer to what might otherwise be a confusing conundrum related to the following:
We’re told that the ostensible reason you can never escape from inside a black hole is that the escape velocity at the event horizon is the speed of light, and of course even greater further in.
But one could argue that, at least in abstract theory, you don’t need to achieve escape velocity to get arbitrarily far from any object such as Earth. If you had some amazing kind of rocket thruster that could generate a little more than 1 G of acceleration for whatever mass the rocket had, and could continue to do it indefinitely, you would rise slowly from the earth’s surface, eventually reaching outer space, and as you got farther and farther away and the thrust continued at the same level, as the earth’s gravity weakened the rate of acceleration would increase and there you go – you’d eventually be on a no-return trajectory. The fact that we could probably never build such a rocket engine is irrelevant – it doesn’t violate any basic laws of physics.
IOW, escape velocity is just a parameter that applies to unpowered ballistic projectiles – how fast would it have be moving when departing the Earth in order to never return. This is an extremely important parameter in practice but not at all in this little thought experiment.
So why, then, could we not take such an implausible but theoretically possible rocket to a supermassive black hole (one with a non-destructive gravitational gradient at the event horizon) pop into the black hole, look around, then fire up the thrusters to maximum and pop right out again?
ISTM that citing the escape velocity as being light speed doesn’t clearly express what the real issue is. The reason you can’t do the above is because of how completely distorted the geometry of spacetime is at the event horizon. Space becomes so curved that it turns in on itself, and the flow of time as observed from an external frame of reference slows to zero. Beyond the event horizon, a reasonable interpretation of the Einstein field equations suggests that time and space actually switch places. IOW, the reason you can’t fire your thrusters and pop right back out of the black hole is because “out” is no longer a spatial direction, it has become timelike. It’s now your past. There is no “out”.
The main thing about gravity is that it attracts its own “charge”. So two masses attract each other and eventually form a larger mass with even more gravity.
Electromagnetism in contrast attracts its opposite charge. So anything with a net charge will attract matter that serves to neutralize it. Therefore almost everything in the universe is electrically neutral (except at the very smallest scales, where quantum mechanics dominates), and it’s only small residual effects that manifest.
There does seem to be only a single gravitational “charge”, but even if there were two (more more), it wouldn’t change the broad picture much. Like charges would still attract each other. And a mass of like charges would expel any unlike ones, so there would be no neutralization effect as with electromagnetism. Maybe we only see a single charge because our entire (visible) universe has already expelled any other ones (and those ones have collected in some other region of the non-visible universe).
