Calculating the Raleigh Criterion constant to 99 significant figures

[Jenab2]

I set about trying to use HiPER Calc Pro on my phone to solve the integral for the Bessel function of the first kind and of order one, so that I could get the ordinate value for the first root of the function to 99 significant figures, then divide that by π to 99 significant figures, in order to get the Raleigh criterion constant to 99 significant figures. But then I discovered that the HiPER Calc Pro won't do the integration.

In general,

Jɴ(x) = (1/π) ∫(0,π) cos[Nt − x sin t] dt − [sin(Nπ)/π] ∫(0,∞) exp[−x sinh t − Nt] dt

However, for any integer value of N, sin(Nπ) = 0.

And so, when N is an integer, you can just solve the former term,

Jɴ(x) = (1/π) ∫(0,π) cos[Nt − x sin t] dt

And, in this case, it happens that N=1. Then you find the least positive value for x for which J₁(x) = 0.

The HP Prime G2 will solve this integral, and, to 12 significant figures, the Raleigh criterion constant is 1.21966989127. But that's as much precision as I can get from the HP Prime G2.

The Raleigh criterion (or Dawes limit) is the minimum angular size or separation, θᵣ , that can be resolved by a telescope having a circular aperture of diameter D, at wavelength λ, where D and λ have the same length units. The equation for the Raleigh criterion is

sin θᵣ = 1.21966989127 λ/D

I found another way to solve the Bessel function of the first kind for integer orders.

Jɴ(x) = Σ(k=0,∞) (−1)ᵏ (x/2)ᴺ⁺²ᵏ / [(N+k)! k!]

and

J₁(3.831705970207512315614435886308160766564545274287801928762298989918839309519011470214112874757423127) = 0.

That argument, divided by π, is the Raleigh criterion constant to 99 significant digits:

1.21966989126650445492653884746525517787935933077511212945638126557694328028076014425087191879391333

 

[jedishrfu]

This is probably an issue with hiper calc and its internal 100 digit limitation for any number. If instead you had wanted say 50 digit precision it probably would work.

 

[tech99]

The correct spelling is Rayleigh. There is an article about him here:-
https://www.nobelprize.org/prizes/physics/1904/strutt/biographical/
I don't think there is any actual need for precision with this constant.

 

[sophiecentaur]

I don't think there is any actual need for precision with this constant.

You are absolutely right. The Raylieigh Criterion is very much an arbitrary rule of thumb for predicting the resolving power of a lens system (it assumes circulat symmetry and a flat field etc etc.). It basically tells you when the dip in brightness patterns of two (equally bright) point sources is half the power of the two maxima. that is considered to be the 'best you can do' but we all know that, with a bit of care (plus some number crunching), you can do a lot better than that. The real limit is down to the brightnesses of the two sources relative to the background brightness (i.e. signal to noise) so a couple of sig figs should be enough.