What is the minimum size telescope needed to resolve this binary?
CLayton asked:
An observer is using a telescope to resolve a binary star with a separation of 0.40 arcseconds using a filter which passes light at 5000 A. What is the minimum size telescope needed to resolve the binary?
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An observer is using a telescope to resolve a binary star with a separation of 0.40 arcseconds using a filter which passes light at 5000 A. What is the minimum size telescope needed to resolve the binary?
Steps would be much appreciated.
December 21st, 2009 at 9:42 pm
The Raleigh Criterion for angular resolution is probably the limiting factor here.
sin(theta)=1.22*(wavelength)/(diameter of aperture)
Rerranging:
Diameter=1.22*(wavelength)/(sin(theta))
=1.22*(5000e-10 m)/(sin(1.94e-6 radians))
=.31 meters in diameter Miss Fantastic
December 25th, 2009 at 6:41 am
Assuming that the telescope is of high quality construction, and that the atmospheric conditions are perfect for observations, then the resolution is limited by diffraction.
Light passing through the lens interferes with itself creating a ring-shaped diffraction pattern, known as the Airy pattern, if the wavefront of the transmitted light is taken to be spherical or plane over the exit aperture. The result is a blurring of the image. An empirical diffraction limit is given by the Rayleigh criterion invented by Lord Rayleigh:
The images of two different points are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other. If the distance is greater, the two points are well resolved and if it is smaller, they are not resolved. Mathematically, this translates into:
D=1.220 x w/sin r
Where D is the diameter of the main mirror or lens
w is the wavelength of the light
r is the resolution required in radians
In this case we have:
w = 5000 Angstroms
r = 0.4 arc seconds = 0.000111111 degrees = 1.93925 x 10^-06 radians
sin r = 1.93925 x 10^-06 radians
Therefore
D= 1.220 x 5000 / 1.93925 x 10^-06 Angstroms
D = 3,145,538,295 Angstroms
D = 31.46 centimeters
or about 12 inches. Mountainboy19682