Ask Ethan: Why are astronauts weightless while still in Earth’s gravity?

If you stand here on the surface of the Earth, whether at the north pole where the acceleration is greatest or at the summit of Huascarán where it’s the smallest, you’ll be pulled towards the center of the Earth with an acceleration of 9.8 m/s²: the acceleration due to gravity, or “1g” on the surface of the Earth. Sure, there are tiny variations that will ensue, as your distance from Earth’s center, the speed of the planet’s rotation beneath your feet, and the density and composition of Earth’s crust and mantle aren’t constant, but those variations only lead to variations in the gravitational acceleration at the less-than-1%-level.

That’s why, if you bring a scale to any of those locations on Earth’s surface and stand on it, you’ll find that the reading on that scale gives you your weight from a very simple formula: your mass (m) multiplied by the acceleration of gravity (1g) on Earth’s surface. If you have a mass of 100 kg, that means your weight will be 220.5 pounds, or 980 newtons, depending on which measuring system you use. But for an astronaut in space, whether on the International Space Station or on their way to or from the Moon, they’re completely weightless: their weight is zero. But why is that? That’s what Graeme Cree wants to know, writing in to ask:

“Gravity reduces in force the farther you get from the source, of course. If the attraction is ‘X’ at one location, and you double the distance from the source, the attraction will only be ‘¼X’.
Okay, so my understanding is that you measure the earth’s attraction from the core of the earth. At Sea Level I’d be about 4,000 miles from the core. So, if my weight is 160 at Sea Level, and I moved another 4,000 miles farther away from the Core, my weight should be 40. But in that case, why are we weightless in orbit, only a hundred miles or so from sea level?”

This simple-sounding question is actually quite profound, and gets at the heart of what the relationship between acceleration and weight actually are. Let’s figure it out together.

Wherever you are on Earth, it’s true: gravity always accelerates you downwards, towards the center of the Earth, at a rate of 1g: 9.8 m/s². However, you don’t always actually accelerate at that rate! In fact, you’re probably not accelerating downward at that rate right now. If you’re sitting in a chair, standing with your feet on the ground, or laying down on a bed or couch while reading this, your acceleration is probably 0 m/s² instead. You’re not speeding up in your motion, and you aren’t “falling” closer towards the Earth’s center at the moment. Instead, you’re remaining exactly where you are.

But you can actively “feel” your weight as well. For example:

• if you’re sitting, you can feel your weight through the seat you’re sitting in, where the chair pushes up on you,
• if you’re standing, you can feel your weight through your feet, as the ground pushes up on your shoes, preventing you from falling through it,
• and if you’re laying down, you can feel the surface you’re laying on pushing back up on your body.

In other words, even though there is the force of gravity acting on you — the gravitational force, after all, is indeed universal — because there’s a solid surface beneath you pushing you back up, your acceleration is zero.

In Newtonian terms, the reason you aren’t accelerating is because there’s an equal-and-opposite force pushing back on you against the force of gravity: something that we call the normal force. (Normal, because in mathematical terms, it’s “normal,” or perpendicular, to the flat surface itself.) On a solid, flat surface, the force of gravity may be pulling you down, but the force of the surface itself pushes back up on you, equally and oppositely, so that there’s a zero net force acting on you overall. That’s why you aren’t accelerating, and why you don’t get closer to the Earth’s center over time.

But there is a way to remove that normal force entirely: you can jump or fall. As soon as you lose contact with whatever surface was holding you up — removing the normal force entirely — gravity immediately begins accelerating you downward at 9.8 m/s². This happens:

• when you jump down from a height to the ground below,
• when you jump straight up-and-down from a spot on the ground,
• when you skydive by jumping out of an airplane,
• or when you swing on a swingset and jump out of your seat.

In all of those cases, as long as you’re in free-fall, there’s still gravity acting on you, but, albeit only briefly, you’re effectively weightless.

There’s a feeling we associate with that weightlessness: the feeling of your stomach rising up into your chest, bringing a wave of nausea and discomfort, often akin to the “butterflies in your stomach” feeling. You don’t just feel it when you fall or jump, but also when you ride a roller coaster, when you ride in a car that moves too quickly over a crest in an uneven road, or — as shown above — when you’re a passenger in a plane that suddenly cuts its engines, and enters into “free fall” for several seconds. NASA’s astronaut trainer aircraft, colloquially known as the vomit comet, does exactly this.

There’s a physiological reason for this. When you’re at rest on the ground, whether sitting, standing, or reclining, gravity pulls down not just on your body, but on all of your organs within your body. The equilibrium that you achieve when you’re stable and not accelerating on the ground arises from everything within your body being pulled down, towards the center of the Earth, by the force of gravity, and by the adjacent, “underneath” part of your body pushing back up on it. What you perceive as your equilibrium configuration is your body’s configuration with Earth’s gravity pulling down on all of its components, and with the normal, restoring force of the solid Earth beneath you (as well as whatever’s atop it) pushing back up.

If you continued to stack objects up — and up, and up, and up — until you were extremely far away from the center of the Earth, like hundreds or even thousands of kilometers up, what would you feel as you stood atop that giant planetary megastructure?

As long as there is still a normal, restoring force that prevents you from entering that free-fall state — where you’d begin accelerating downwards at the acceleration due to gravity at your location — things would be much the same as they are for you on the surface of Earth today. The only major difference is that, since you’re farther away from Earth’s center and because gravity is an inverse-square force law (where the force is proportional to the separation distance between the two masses in question squared), the effective force of gravity, and hence the effective acceleration due to gravity, is reduced.

If you were:

• 100 km up (at the Earth’s upper sodium layer), your acceleration would be 9.5 m/s²,
• 400 km up (near the ISS), your acceleration would be 8.7 m/s²,
• 600 km up (like a Starlink satellite), your acceleration would be 8.2 m/s²,
• 1000 km up (just below the inner Van Allen belt) your acceleration would be 7.3 m/s²,
• 2000 km up (into medium-Earth orbit), your acceleration would be 5.2 m/s²,
• or 6371 km up (at one Earth-radius from Earth’s surface), your acceleration would be 2.4 m/s².

Effectively, your weight would be reduced to your new acceleration divided by Earth’s acceleration at the surface.

However, standing on a platform that’s attached to Earth’s surface is a very different scenario than actually being in orbit, or in free-fall, at that same altitude. One way to conceive of this is to remember that Earth rotates once in a 24 hour period, and that at the equator, the radius of the Earth is around 6378 km. At Earth’s surface, a mass standing on the equator moves with a speed of around 1670 km/hr (1040 mph), and so adding extra altitude simply equates to being farther from Earth’s center and tracing out a slightly larger circle as you move along with the planet.

For the same altitudes we just considered:

• 100 km equates to a speed of 1696 km/hr,
• 400 km equates to a speed of 1775 km/hr,
• 600 km equates to a speed of 1827 km/hr,
• 1000 km equates to a speed of 1932 km/hr,
• 2000 km equates to a speed of 2194 km/hr,
• and 6371 km equates to a speed of 3340 km/hr.

Those speeds are somewhat faster than a person moves on Earth when they stand at Earth’s equator, but not by an extraordinary amount. However, all of those speeds are extremely slow compared to the actual speeds of satellites that orbit the Earth. If we consider an astronaut on the International Space Station, for example, at their altitude of 400 km, they move at approximately 27,600 km/hr, or more than ten times as fast as someone standing on a platform connected to Earth at the same altitude would move at.

A satellite is a very counterintuitive thing for most of us. We normally view gravity as an irresistible force: something that pulls us back down no matter how far away we travel away from the confining mass. But because it moves so quickly, a satellite is basically the equivalent of “throwing a mass at the Earth and missing.” The only way you can do that is by having your mass move at such a high speed that Earth’s gravity — strong though it may be — is insufficient to actually bring that mass back down to collide with Earth’s surface. It’s the exception to “what goes up must come down,” and the way you create that exception is with enough speed (or, equivalently, kinetic energy).

In the case of a satellite, you’re in motion that’s so rapid, even relative to the planetary mass that you’re orbiting, that the acceleration due to gravity pulls you into an elliptical orbit, but nowhere does that orbit actually intersect with the surface of the planet itself. Just as the Moon is in free-fall around the Earth, so too is an Earth-orbiting satellite. Even though it’s continuously being accelerated by the gravity of Earth towards Earth’s center, the fact that it’s moving at such a high speed ensures that it will remain in a stable orbit around our planet. As a result, a satellite, and everything on it, is still in free-fall.

Another way to think about this is to consider the object at the core of our entire Solar System: the Sun. The Sun is much, much more massive than the Earth is: over 300,000 times as massive. The Earth, and everything on it, and everything orbiting our own planet, also orbits the Sun. The Earth moves much more quickly around the Sun than satellites do around the Earth: at a speed of around 107,000 km/hr, or 67,000 mph.

On an even grander scale, the Sun and the entire Solar System orbit around the center of the Milky Way, at a monstrous distance of 26,000 light-years from the galactic center. All told, those speeds even dwarf the Earth’s motion around the Sun, as our entire Solar System travels at approximately 790,000 km/hr (490,000 mph) with respect to the Milky Way’s center.

And there’s an even greater speed: the Milky Way galaxy moves within our Local Group of galaxies, and our Local Group moves relative to the Hubble flow of the Universe at a whopping 2.26 million km/hr (1.4 million mph). But, of course, we don’t feel it, and the reason is the principle of relativity itself. You don’t experience absolute motion, only motion relative to some other frame of reference.

The reason astronauts are weightless is because they aren’t connected to Earth, and there isn’t a normal force from any solid surface pushing back on them. They may be accelerating relative to Earth, but the spacecraft that they’re inside is also accelerating at the same rate as they are: with the same magnitude and in the same direction. As a result, there’s no relative acceleration between the astronauts and the spacecraft that they’re inside. It’s the same situation, physically, as:

• being an astronaut inside the vomit comet,
• being a human in an elevator that has the cables cut and begins to descend,
• being a skydiver who initially jumps out of an airplane (before they pick up speed and air resistance becomes significant),
• or a passenger in a Drop Tower amusement park ride.

It’s also the same situation as an astronaut being completely outside of a spacecraft, freely floating (or freely falling) in outer space, being accelerated only by whatever gravitational forces are influencing them. So long as there’s nothing pushing or thrusting the astronaut themselves — nothing that will cause them to feel an acceleration relative to their spacesuits — they aren’t going to “feel” their acceleration. That doesn’t mean that they aren’t accelerating, but it means that they won’t feel it as weight.

If we return to the original situation — where you’re stationary on the surface of the Earth — one of the most widespread myths that we encounter is that our weight is a property intrinsic to ourselves: that it’s equal to our mass multiplied by the acceleration due to gravity. (You can find this myth in many introductory physics textbooks!) But it’s only your mass that’s intrinsic to you; your weight is what’s determined by the normal force pushing back on you: the force that whatever surface you’re on that’s keeping you from falling, whether the floor, a seat, a couch, a bed, etc., exerts on you.

If you get into an elevator and stand on a scale, when the elevator accelerates upwards, you’ll see your weight rise, as that upward acceleration causes the normal force on your feet to be greater than your mass multiplied by the acceleration due to gravity. (That’s good! You need a positive, upward net force in order to accelerate you upward!) Similarly, when the elevator accelerates downward, you’ll see your weight drop, as a downward acceleration results in a less-than-usual normal force. If you made it this far, congratulations! You now understand weight better than almost everyone else, including many physics teachers. In the absence of a normal force, everything is weightless, regardless of your mass or your acceleration.

Send in your Ask Ethan questions to startswithabang at gmail dot com!

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