Accelerometer in space

Question

If my understanding of the General Theory of Relativity is correct, according to the Equivalence Principle, forces due to gravity and acceleration are indistinguishable.

If that's the case, then an accelerometer on Earth's surface would always read "acceleration" of $9.8 \frac{\text{m}}{\text{s}^2}$ in vertical direction due to Earth's gravity. Is my train of thought correct, or am I completely out to lunch?

OK, moving on. Now, if I wanted to devise a very precise INS (inertial navigation system) for interplanetary travel, I would need to constantly account for "false acceleration" due to gravity of nearby (or all?) planets and their moons within the Solar System. Is that right?

Could someone please explain?

NB: I am aware of error accumulation inherent in INS, but this is not what this question is about.

Answer 1

Yes, a properly functioning accelerometer that is stationary relative to the surface of the Earth will read the acceleration due to gravity. If it's a very good accelerometer, you could also see the factor of a few hundred smaller decrement in that acceleration due to the centripetal acceleration from the rotation of the Earth. If it's a really really good accelerometer, you could also see the tidal force due to the Moon, about seven orders of magnitude down from the surface acceleration.

An accelerometer in the spacecraft, at least if it's at or close enough to the center of gravity of the spacecraft, will read exactly zero if the spacecraft is in free fall. I.e., not firing any thrusters. The accelerometer is useful during rocket firings to get a measure of the actual acceleration during the burn. (This is usually improved upon later with navigational tracking.) But useless in free fall.

You will not detect the gravity of nearby objects with an accelerometer. It will read zero the entire time you are moving through the vacuum. With one caveat, which is again if it's a really really good accelerometer, and you swing by very close to a large body, and you put the accelerometer as far from the center of gravity of the spacecraft as you can, then you might be able to detect the tidal force from the body for a short time. That force drops off as the cube of distance as opposed to the square, so it would be very difficult to detect. In any case, you would not be able to navigate with that, since if you're already that close to the body then it's too late to correct your trajectory.

In general, you cannot navigate a spacecraft, e.g. correcting errors in your trajectory, without, effectively, looking out the window. Either with optical or radio tracking data of some kind.

Answer 2

If that's the case, then an accelerometer on Earth's surface would always read "acceleration" of 9.8m/s^2 in vertical direction due to Earth's gravity. Is my train of thought correct, or am I completely out to lunch?

You are correct, but only to two decimal places of accuracy. An accelerometer that gives only two decimal places of accuracy is an extremely lousy accelerometer.

At sea level and at the equator, acceleration due to gravity is 9.8142 m/s$^2$, directed toward the center of the Earth. An accelerometer will register an acceleration of only 9.7803 m/s$^2$, directed upward. These differ in magnitude by 0.0339 m/s$^2$, or about 0.34%. (Note well: They differ in direction by 180 degrees.) Even a lousy accelerometer is good to three places of accuracy, so this variation of 0.34% is within the realm of detection by a mediocre accelerometer (but not an extremely lousy accelerometer).

From a Newtonian perspective, an ideal accelerometer measures the acceleration due to the net sum of all real forces except gravity. What about fictitious forces such as the centrifugal and Coriolis forces? They don't count from a Newtonian perspective. They're fictitious forces. Ideal accelerometers don't sense those observer-dependent fictitious forces. What about gravity (which is a real force in Newtonian mechanics)? Accelerometers don't detect that, either. That's the only real force accelerometers don't detect from a Newtonian perspective.

From a relativistic perspective, an ideal accelerometer measures proper acceleration. Gravitation is a fictitious force in general relativity. A simple way to put this: An ideal accelerometer measures the acceleration due to the net sum of all real forces, period. Another way to put this: accelerometers measure all real forces except gravitation (the same explanation as the Newtonian explanation).

There are some subtleties that the above ignores. @Rikki-Tikki-Tavi mentioned frame dragging in a comment. Unless you are orbiting a rotating black hole, frame dragging is exceptionally small. One would need an extremely sensitive accelerometer to sense frame dragging, and even then only after a whole lot of analysis. Gravity Probe B was intended to measure those relativistic effects. This experiment was at best a partial success. The subtleties to which @Rikki-Tikki-Tavi alluded are very, very small for the Earth.

Now, if I wanted to devise a very precise INS (inertial navigation system) for interplanetary travel, I would need to constantly account for "false acceleration" due to gravity of nearby (or all?) planets and their moons within the solar system. Is that right?

Since accelerometers do not detect gravitation, deducing the evolution of a spacecrafts state (position and velocity) means that a spacecraft needs decent gravity models of the objects that affect the spacecraft gravitationally. This is called deduced reckoning, or dead reckoning for short.

There's a good reason for the change from ded(uced) to dead. A spacecraft that only relies on dead reckoning alone is soon to be dead.

I mentioned "ideal accelerometers" multiple times above. Real accelerometers differ from the ideal in a number of ways. A real accelerometer might report an acceleration of 1.001 m/s$^2$ when it should have reported an acceleration of 1 m/s$^2$, and an acceleration of 2.002 m/s$^2$ when it should have reported an acceleration of 2 m/s$^2$. This is a scale factor error.

Take an accelerometer apart and you'll see multiple devices that ideally measure acceleration in three orthogonal directions. In practice, they don't. Real accelerometers have a non-orthogonality error.

Scale factor and non-orthogonality are examples of the systematic errors associated with an accelerometer. Systematic errors can be addressed. A decent navigation system will try to estimate these systematic errors.

What can't be addressed are random errors. Accelerometers are inherently noisy devices, ideally white noise. Integrated white noise results in a random walk. The deduced velocity vector takes a random walk from the true velocity vector. Flight software computes position by integrating velocity. This results in an integrated random walk error.

Over time, this can result in very bad estimates of position. That's why dead reckoning means you are dead. The only way to overcome this is to have an alternate measure of position. GPS does a very nice job for vehicles in low Earth orbit. Something else is needed for vehicles beyond LEO.

Answer 3

There's already an answer, but to phrase it in a different way:

Accelerometers always measure acceleration with respect to a "body" frame, usually the body of the accelerometer. On the ground, the accelerometer is virtually fixed in place, so any accelerations show up unchanged. On a spacecraft, the frame is accelerating (because the spacecraft is in orbit/free fall), so the difference in acceleration is very small.

Answer 4

I found this description on Wikipedia. An accelerometer is a device that measures proper acceleration ("g-force"). Proper acceleration is not the same as coordinate acceleration (rate of change of velocity). For example, an accelerometer at rest on the surface of the Earth will measure an acceleration g= 9.81 m/s2 straight upwards. By contrast, accelerometers in free fall orbiting and accelerating due to the gravity of Earth will measure zero.

This was basically mentioned in the answers above. However, I would like to add that if we are talking about a single sensor accelerometer, and the accelerometer rests on the earth at an angle it will read less than 9.81m/s2. For example, if the accelerometer rests at a 45 deg. angle the reading will be 9.81 x cosine 45 which is roughly 9.81 x 0.7. Lesser angles will read less acceleration. And when the accelerometer rests parallel to the earth surface, it will read zero. I can see where accelerometers can present problems in certain situation, and corrections will have to be calculated.

If a rocket is going into an earth orbit, the effect of gravity on the accelerometer will lessen as the rocket is making it's turn from a vertical position to the horizontal position to the earth surface while still accelerating. A 3-axis accelerometer can overcome the directional problems, however gravity will still have an effect on the reading and must be taken into consideration.