# No uncertainty for standard gravitational acceleration?

The other day I asked about the uncertainty of light, and this issue triggered me to start looking into other physical constants and try to understand why other constants have no uncertainty.

One of those constants is the standard gravitational acceleration, which is the acceleration due to Earth's gravity ($\approx 9.8\: \mathrm{m/s^2}$)... which is also a physical quantity for which I could easily suggest a lot of reasons why it should have uncertainties, like:

• The meter itself has uncertainty,
• The radius of the Earth is different everywhere

Why doesn't $g$ have uncertainty?

I would like to point out there are other constants with no uncertainty (like permittivity of free space) that I don't understand. Isn't there a collective reference to explain those conventions?

EDIT:

Maybe I wasn't clear. Check the latest review from Physical Review D, and you'll see that the uncertainty is given as zero at page 108. And also from CODATA NIST:

The typical gravitational acceleration on the surface of the Earth, $g \approx 9.8\: \mathrm{m/s^2}$, has uncertainty. That's one of the reasons why the $\approx$ symbol is used.

The Earth's gravitational field varies a lot due to oceans, the thickness of the crust, mountains, non-uniform density in the crust and mantel, etc.

A pair of satellites was launched for the Gravity Recovery and Climate Experiment (GRACE) and based on that data a map was made showing the variation: From Wikipedia:

Apparent gravity on the earth's surface varies by around $0.7\%$, from $9.7639\: \mathrm{m/s^2}$ on the Nevado Huascarán mountain in Peru to $9.8337\: \mathrm{m/s^2}$ at the surface of the Arctic Ocean.

As others have mentioned, the constant of gravitational acceleration, $g_0$ that is defined exactly as $9.80665\: \mathrm{m/s^2}$ is used for the standardization of weight like the pound against units of mass like the kilogram.

• I should mention that often the word "uncertainty" is used for things that are constant but can't be measured to infinite precision. Since $g$ varies, you don't see a lot of mention of the uncertainty of $g$ but rather the variation of $g$. GRACE also shows $g$ at a specific location varies over time. – Brandon Enright Jan 12 '14 at 4:23
• Reading about Earth's gravity takes you on several unintuitive turns. This answer gives the correct mechanism, inhomogeneity, for the majority of the variation in $g$. Yet, this wasn't even on the list of the OP's expectations. It's also surprising that gravity increases with depth, when the simple model predicts that it should decrease. The reason here again is inhomogeneity. But hey, if you're working off the mathematical framing, we would probably be a "regolith anomaly" ourselves. – Alan Rominger Jan 12 '14 at 21:24
• @AlanSE it's also amazing to see that convection in the mantle actually changes the gravitational acceleration in a location over time. Naive assumptions would suggest that for a given depth, the mantle is mostly uniform in heat, density, and material. – Brandon Enright Jan 12 '14 at 21:28

You're actually looking at the "standard gravitational acceleration," $g$, which is not the same thing as the actual acceleration of gravity. This constant $g$ is used to provide a consistent definition of certain units of force, for example the pound force (lbf), so it needs to be a defined constant, not a measurement. $g$ was chosen to be close to the acceleration of gravity on the Earth's surface (for obvious reasons), but other than that it has nothing to do with the actual strength of the Earth's gravitational field.

In particular, $g$ does not reflect the most up-to-date measurements of the strength of Earth's gravity. Don't let its appearance in the PDG Review (your PRD reference) fool you: that number has been the same for decades. If you go back and check it in previous versions of the review, you will find the same value.

The table cited is giving the definition of a unit called the standard acceleration of gravity. It happens that the acceleration of gravity on various places on earth is close to 1 when measured in these units, but that does not change the exactness of the definition.

By analogy. One foot is exactly 12 inches, despite the varying size of various human feet found in the population...