when is radian day?

[buhrger]

both π day and the more recent τ day (both more recent = it was yesterday and more recent = it hasn't been celebrated for as many years) suffer from the same problem: they depend on the decimal expansion of one or t'other of the circle constants. using base 10 is precisely the sort of arbitrary choice that going to radian measure of angles is attempting to avoid. thus, one wonders, is there a more appropriate way to select a day to celebrate all things circular?

one obvious choice would be to celebrate when the earth has moved through one radian of its orbit, i.e., when the distance it's travelled (from, e.g., perihelion) is equal to its radius from the sun. there are far fewer mathematically arbitrary choices involved in this. perihelion is around january 4. and 365.242199/τ (365.242199/(2π) for the π-ous) is 58.130101389 = 58 days + 3 hours + 7 minutes + 20.76 seconds. doing the math suggests that radian day would be around march 4th, which conveniently is also GM's day.

two problems:

1. someone else thought of it first. technically, this is the opposite of a problem. although, by starting the radian-clock rom perihelion (i suppose one also might want to use winter solstice?), my radian-day ends up on a slightly different date.
   
   
2. this one, however, is a problem: the earth's orbit famously ain't a circle. thank you, johannes kepler! it's an ellipse. and calculating lengths along an ellipse will require futzing about with, well, elliptic integrals (guess why they're called that). maybe later.