Blurring of an Object by a optical/radio Telescope
Joachim Köppen Kiel July 2019

### Some brief explanations

• Measuring the apparent (angular) size of an object (in the sky) depends on the angular resolution of the telescope or instrument, i.e. the width (FWHM) of the instrumental profile. In a radio telescope this is the width of the main lobe (HPBW = half power beam width).
• The simulation can be done in one or two spatial dimensions. Several simple intensity profiles for the object are given. Note that the double sources can be done in 1D only.
• Is the telescope's FWHM much smaller than the size of the object, the apparent angular width is very close to the true size of the object. But if it is much larger, the apparent angular width is very close to the instrument's FWHM. If both object and instrument have Gaussian profiles, the resulting FWHM is the root of the sum of their squares. This convenient formula may serve as a good approximation for other profiles, too.
• For a circular (2D) object, we also give the filling factor simply guessed from the ratio of the widths: MIN(1, (FWHMobj/HPBW)2) is a rather rough formula. For Gaussian profiles the expression FWHM²obj/(FWHM²obj+HPBW²) is exact, and may also be used as a good approximation for other profiles. The true filling factor is computed from the true and apparent intensity profiles. Note that because of the limited numerical accuracy this value is no longer reliable below about 0.005 and therefore is not displayed.
• The initial example is the Sun observed with a radio telescope with a 0.5 deg beam width.

The object

object FWHM
object separation
L/R intensity ratio

The antenna/instrument

antenna FWHM

The result
apparent FWHM
obj+F²ant
filling factor guesses:
min(1,(Fobj/Fant)²)
obj/(F²obj+F²ant)
true value (2D only)