# Which is the “roundest man-made object”: Gravity probe B rotor or the Avogadro project sphere?

The Avogadro project is part of the efforts to replace the artefact-based standard kilogram with a definition based on naturally reproducible objects. The main challenge is to make a very precise sphere of isotopicaly pure silicon. There are often-repeated popular claims that the silicon sphere from the Avogadro project is the "roundest object ever made by man".

However, the same title is ascribed to rotors of Gravity Probe B's gyroscopes, even earning then a certificate from Guinness book of World Records.

So, which one is rounder (in appropriately defined way) -- the silicon sphere or the Gravity Probe B rotor?

-

On the one hand, the caption under the image of the sphere being held by one of the member of The Avogadro Project reads as follows:

The roundness delta of the finished sphere (being held above) is about 50 nm on a 93.6 mm diameter. It is believed to be the roundest object in the world.

So that its departure from mathematically perfect sphericity is $$\frac{5 \cdot 10^{-8} \text{m}}{93.6 \cdot 10^{-3} \text{m}} \approx 0.053 \cdot 10^{-5} = 5.3 \cdot 10^{-7}$$ of its diameter.

On the other hand, the certificate that lists the GP-B gyroscopes in the Guinness Database of World Records as "the most spherical man-made objects" reads as follows:

The most spherical man-made objects are the fused quartz gyroscopic rotors onboard the Gravity Probe B Spacecraft operated by NASA and Stanford University. Their average departure from mathematically perfect sphericity is only $1.8 \cdot 10^{-7}$ of their diameter.

Therefore, the GP-B gyroscopes are rounder than the Avogadro Project's sphere ($1.8 \cdot 10^{-7}$ vs. $5.3 \cdot 10^{-7}$) but - considering that the first ones are on a satellite - maybe the Avogadro Project team is not lying by claiming that its silicon sphere is "the roundest [man-made?] object in the world".

-
Thank you for the detailed analysis and the nice resolusion of the conundrum -- let the silicon sphere be the roundest on Earth! –  Slaviks Jan 12 at 17:27
I worked on the Gravity Probe B mission. We did have spares and they are still on earth, technically. –  user55643 Jul 17 at 15:55

Gravity Probe B had four gyros, so one would expect that they made some spare rotors. There is a notation on the Gravity Probe B Website that Gyro #4 was replaced before launch. So very likely there is a spare Gravity Probe B rotor sitting in a box somewhere that is the roundest object on Earth.

BTW the coefficient of thermal expansion of silicon is 2.6e-6 per degree C so unless you keep the sphere's temperature really uniform then it will develop thermal asymmetries.

-