A245461 Decimal expansion of constant in the Rayleigh criterion: first zero of J_1, divided by Pi.

1, 2, 1, 9, 6, 6, 9, 8, 9, 1, 2, 6, 6, 5, 0, 4, 4, 5, 4, 9, 2, 6, 5, 3, 8, 8, 4, 7, 4, 6, 5, 2, 5, 5, 1, 7, 7, 8, 7, 9, 3, 5, 9, 3, 3, 0, 7, 7, 5, 1, 1, 2, 1, 2, 9, 4, 5, 6, 3, 8, 1, 2, 6, 5, 5, 7, 6, 9, 4, 3, 2, 8, 0, 2, 8, 0, 7, 6, 0, 1, 4, 4, 2, 5, 0, 8, 7

Offset: 1

Author

Charles R Greathouse IV, Jul 22 2014

Keywords

• nonn
• cons

Comments

The angular resolution (diffraction limit), in radians, is this constant times the wavelength of the light used divided by the aperture diameter. That is: objects closer than this angle cannot be resolved as separate objects.

Note that the small angle approximation sin(theta) =~ theta can be used. Also, a commonly used approximation to this constant is 1.220. - John W. Nicholson, Jul 29 2014

For example, a radio telescope with a 300-meter aperture observing at the hydrogen line (21.106... cm) can distinguish objects at most 0.04916... degrees apart. - Charles R Greathouse IV, Mar 08 2018

Examples

			1.21966989126650445492653884746525517787935933077511...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• COSMOS - The SAO Encyclopedia of Astronomy, Rayleigh Criterion
• Rene Kay, Todd Kay, Planet Seeker Interferometer (PSI), hal-02870885v2 [astro-ph.IM], 2020.
• Wikipedia, Angular resolution
• Wikipedia, Diffraction-limited system
• Index entries for transcendental numbers

Crossrefs

Cf. A115369, A221210.

Programs

Mathematica

RealDigits[ BesselJZero[1, 1]/Pi, 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2014 *)

PARI

solve(x=3,4,besselj(1,x))/Pi

PARI

besseljzero(1)/Pi \\ Charles R Greathouse IV, Aug 09 2022

Formula

Equals A115369/A000796.