Thursday, May 16, 2019

[zdoktiye] The redefinition of the parsec and the (possibly) new value of pi

Originally, the parsec in astronomy was defined in terms of a right triangle with one vertex having an angle of 1 second of an arc = 1 arcsecond = 1/3600 of a degree.  (The other two vertices had angles 90 degrees and 89+3599/3600 degrees.  It was a very skinny right triangle.)  The ratio of the length of the short leg to the long leg of the triangle was tan(1 arcsecond) = tan(pi/(60 * 60 * 180) radians).  This ratio gave the (old) conversion factor between the astronomical unit and the (original) parsec.

Nowadays, as a result of a 2015 International Astronomical Union resolution (according to Wikipedia), that right triangle has been thrown away, and the conversion factor between au and parsec is defined to be exactly pi/(60 * 60 * 180), which is what one would get from the above tangent expression if one defined tan x = x exactly (where x=pi/(60 * 60 * 180)).  The approximation tan x ~= x is good for small angles but exact only at x=0.

Here are the old and new conversion factors.  We underline where the digits start to differ.

Parsecs per Astronomical Unit
old 0.00000484813681113334... tan(pi/648000)
new 0.00000484813681109535... pi/648000

Reciprocal of above, the more common conversion factor:

Astronomical Units per Parsec
old 206264.8062454803... cot(pi/648000)
new 206264.8062470963... 648000/pi

Another way of looking at it is the parsec got longer by about 0.0000016 au.  The au is defined to be exactly 149597870700 m, so the parsec got longer by about 241756 m (242 km).

Are there any astronomical measurements being made in which the difference between the old and new parsec matters?  Maybe things involving interferometry, e.g., VLBI or LIGO.

The right triangle calculation above assumes space is flat and light travels in straight lines.  Of course, general relativity says space is not flat and light curves, affecting parallax measurements.  How much error does the assumption that space is Euclidean induce in distances measured by parallax?  How does that error compare in scale to the scale of the change in the definition of a parsec?  (What are the most important sources of space-bending with regards to the parallax?  Earth, Sun, Sagittarius A*, Dark Energy?)

We might also have light bending due to refraction caused by the varying densities of the not quite perfect vacuum of space.

Just for fun, we could keep the right triangle but redefine the angle that defines the parsec: no longer 1 arcsecond but arctan(pi/(60 * 60 * 180)) radians or 0.9999999999921651... arcsecond, or 00.59 59 59 59 59 59 38 04 03 09 53... arcsecond in base 60.

Incidentally the Taylor series (Maclaurin series) for arctangent converges quickly for these small angles, and is useful for computing the approximate difference between the exact and small angle approximation.

And finally just for fun, we could keep the right triangle and its vertex angle of exactly 1 arcsecond, but redefine pi, specifically the pi used to convert from degrees (1/3600 of a degree in our case) to radians.  Let MP mean old pi, the mathematicians' traditional value of pi.  The new conversion factor stays the same (new) value given above, 0.00000484813681109535... pc/au, now calculated as MP/(60 * 60 * 180).  Then, we want a value of pi such that

tan(pi/(60 * 60 * 180)) = MP/(60 * 60 * 180)

We solve for pi, and get pi = arctan(MP/648000) * 648000, or

pi = 3.141592653565179456360176669261077433...

We have underlined where the digits start to differ from the old value of pi.  The new value value of pi is smaller by about one part in 128 billion (1 part in 128 * 10^9), or 0.00000000078%.

The IAU's new definition of the parsec leading to a change in the value of pi is very reminiscent of Indiana defining pi to be 3.2 by fiat, which also was a result of a geometric construction that only "worked" when pi had the value 3.2.

First the IAU came for Pluto.  Then they came for pi.

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