Q:

How are the relative distances of celestial objects from the Earth calculated using observations at a single time instant?

How does one find the distances of celestial objects in the night sky, such as the Moon and the stars, from the Earth using a snapshot of information (including, say, the intensity and wavelength of light received from the various observed objects and their relative positions) observed in the night sky at a single time instant? Most methods (especially those taught in orbital mechanics classes) are inspired by Gauss's method for determining orbits (and hence, distances of the observed objects from the earth), thus requiring observations at several time instants, or equivalently, position and velocity information at a single time instant.

A:

For very distant objects, their distance from us can be estimated in one snapshot by measuring the redshift in their spectra, knowing the so-called Hubble Constant. This method can be refined somewhat if the type of the object (star, quasar, galaxy, etc.) is known and its spectrum can be accurately gathered. 
For a much closer object whose diameter is known, its distance can be estimated trigonometrically by measuring its angular size with a telescope, in one "snapshot". 
If two cameras are allowed instead of one, then two photos of the same object in the sky shot at the same instant from different locations on earth will yield the distance via a parallax measurement, for objects within our local spot in space. 

