Weight in Jules Verne's From the Earth to the Moon

In the recent weight vs. weightlessness post, there was an interesting comment. Michael Maher asks:

My question is this: What forces did the astronauts experience as they were in transit to the moon? In this case the individuals would feel the force of the rocket accelerating them forward (added to the waning gravitational force of the earth) and they would be pulled to the back of the rocket walls, correct? Conversely, as they approached the moon (before entering orbit) would the moons gravitational pull gradually pull them to the front of the ship if it is stronger than the rockets acceleration? Thanks.

Great question. Great enough to deserve it's own post. Ok, let me give a quick answer. No. As long as both the space craft and the astronauts are in a motion due to a gravitational force, they will be "weightless". It doesn't matter if this is a gravitational force due to the Earth, the moon or a combination of the two.

I will give a more detailed answer, but first a connection to literature.

Weight in From the Earth to the Moon - Jules Verne

In case you aren't familiar with this book, it describes a group of people that make a trip to the moon (as the name implies) written in 1865. What does Jules Verne say about the weight of the voyagers as they travel to the moon?

Here is a short passage from the book (which is available in many formats online at Project Gutenberg). In this part, the three travelers are in the space ship (which is just a giant shell that was shot from a canon) and traveling too the moon. The "neutral point" that they refer to is the point where the gravitational force from the moon and the Earth are the same, but in opposite directions.

Such was their situation; and Barbicane clearly explained the consequences to his traveling companions, which greatly interested them. But how should they know when the projectile had reached this neutral point situated at that distance, especially when neither themselves, nor the objects enclosed in the projectile, would be any longer subject to the laws of weight?

Up to this time, the travelers, while admitting that this action was constantly decreasing, had not yet become sensible to its total absence.

But that day, about eleven o'clock in the morning, Nicholl having accidentally let a glass slip from his hand, the glass, instead of falling, remained suspended in the air.

"Ah!" exclaimed Michel Ardan, "that is rather an amusing piece of natural philosophy."

And immediately divers other objects, firearms and bottles, abandoned to themselves, held themselves up as by enchantment. Diana too, placed in space by Michel, reproduced, but without any trick, the wonderful suspension practiced by Caston and Robert Houdin. Indeed the dog did not seem to know that she was floating in air.

The three adventurous companions were surprised and stupefied, despite their scientific reasonings. They felt themselves being carried into the domain of wonders! they felt that weight was really wanting to their bodies. If they stretched out their arms, they did not attempt to fall. Their heads shook on their shoulders. Their feet no longer clung to the floor of the projectile. They were like drunken men having no stability in themselves.

So in Jules Verne's version, you would feel the net force of gravity. This would be true if the space craft were not accelerating. Still, I think it is interesting to see an explanation from quite some time before this was actually accomplished.

Why would you still feel weightless?

Now for an explanation. In my previous post on weightlessness, I showed that you can feel different ways (heavy or light) in cases where the gravitational force doesn't even change. Here is a diagram of a person in an elevator that is accelerating upwards.

So, you don't really "feel" the gravitational force. In fact, you can be in a location with essentially no gravitational force but feel like you are on Earth. This would happen if the elevator you were in (I guess it would be a space elevator) was accelerating such that the floor had to push on you with a magnitude the same as on Earth.

Now, suppose you have rockets on your spacecraft. In that case, if you fire the rockets you will feel a force. Here is a diagram of the Apollo capsule firing rockets on the way to the moon. I have drawn a force diagram for the person inside.

In order for the astronaut to have a motion different than due just to the two gravitational forces, the floor must also push on him. This is what he feels.

Is the "neutral point" the same thing as a Lagrange point?

Ok, back to From the Earth to the Moon. What about this "neutral point"? Here is another passage from the book.

But, in reality, a time must come when the projectile would no longer be subject to the law of weight, after allowing for the other celestial bodies whose effect could not be set down as zero. Indeed, the projectile's course was being traced between the earth and the moon. As it distanced the earth, the terrestrial attraction diminished: but the lunar attraction rose in proportion. There must come a point where these two attractions would neutralize each other: the projectile would possess weight no longer. If the moon's and the earth's densities had been equal, this point would have been at an equal distance between the two orbs. But taking the different densities into consideration, it was easy to reckon that this point would be situated at 47/60ths of the whole journey, i.e., at 78,514 leagues from the earth. At this point, a body having no principle of speed or displacement in itself, would remain immovable forever, being attracted equally by both orbs, and not being drawn more toward one than toward the other.

Now if the projectile's impulsive force had been correctly calculated, it would attain this point without speed, having lost all trace of weight, as well as all the objects within it. What would happen then? Three hypotheses presented themselves.

1. Either it would retain a certain amount of motion, and pass the point of equal attraction, and fall upon the moon by virtue of the excess of the lunar attraction over the terrestrial.

2. Or, its speed failing, and unable to reach the point of equal attraction, it would fall upon the moon by virtue of the excess of the lunar attraction over the terrestrial.

3. Or, lastly, animated with sufficient speed to enable it to reach the neutral point, but not sufficient to pass it, it would remain forever suspended in that spot like the pretended tomb of Mahomet, between the zenith and the nadir.

So the neutral point is where the net gravitational force would have a magnitude of zero. How would you calculate the location of this "neutral point"? Diagram time. This is a mostly to scale diagram of the Earth and moon. There is some point (the red dot) that I am drawing (not to scale) forces from the Earth and moon.

Now, if I call the direction from the Earth to the moon the x-direction I can make a plot. Suppose a 1 kg mass moves from the surface of the Earth to the surface of the moon. This is a plot of the net gravitational force.

Here the red line is the gravitational force due to the moon and the green is for the Earth. Really, these are components of the force in the x-direction. That is why they can be positive or negative. The problem is that the forces are quite large on the surfaces of the objects. Let me zoom in some.

Here you can see the net gravitational force, the blue line. The point where the net gravitational force is zero seems to be around a distance of 3.43 x 10⁸ meters from the center of the Earth 0.886 the distance from the Earth to the moon (center to center). This is a different value than the one from Jules Verne - 47/60 or 0.78. Maybe he had different values for the mass of the moon or something.

Here is another error. The travelers claim that if you get to the neutral line with no speed, you would stay there. Alas, this is not quite true. Why? Well, what do they mean by "remain forever suspended in that spot like the pretended tomb of Mahomet"? If they mean stay in the same spot, that wouldn't work. As time moves on, both the Earth and the moon move so that the gravitational forces change and thus the neutral line would move.

Ah, but maybe Verne meant that it would remain in the same spot relative to the Earth and moon. Again, this won't work. Since the Earth and moon both rotate around a common center of mass (which is inside the Earth), the space craft at the neutral line would also have to orbit. If an object moves in a circle, it must accelerate (centripetal acceleration). And how do you accelerate? You need a net force to accelerate. So, if the neutral line is the point of zero net gravitational force, the object would not move in a circle.

You can get an object to be "balanced" by the gravitational forces and stay in the same relative location. But to do this you still need some force. This is called a Lagrangian Point. Let me draw a not-to-scale diagram of the Earth-moon system with an object at the Lagrangian point.

The Lagrangian point will be closer to the Earth than the neutral line so that there is a net force towards the center of the orbit. There are actually 5 points around the Earth-moon system in which the net gravitational force will make an object at that location be stationary relative to the Earth-moon.

Yes, I left out a full derivation of the location of any of the Lagrangian points. Maybe that will be for another post.