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

Question

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.

Answer 1

I don't think it's possible. Imagine taking a photo of a scene consisting of a lot of point sources, through a camera with a very small lens. Each point source will look the same regardless of distance. They don't shade each other, there's no parallax, etc., etc. If you already know how far away the Sun is, you might be able to estimate distances to various planets by the apparent angle between the planet and the sun, and the arc of the sunlit side of the planet, and knowledge of the laws of gravity.

Answer 2

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.

Answer 3

For near planets, we can use radar ranging. Beyond that, we use trigonometric parallax. Hipparcos measured parallaxes out to about 4.5 light years. Gaia is measuring parallaxes to about 25 thousand light years.

Very long base line interferometry can detect distances of radio sources using an array of radio telescopes, by measuring the time difference for a signal to arrive at different telescopes. This goes out to about 13 thousand light years, I think.

Beyond that we measure luminosity distances. If we know, from studies on near stars, that a particular type of star has a precise absolute luminosity, then we can estimate distance from its apparent luminosity.

Further out we use cosmological redshift.

This is the bones of the cosmological distance ladder. More detail is in Structures of the Sky, and in many other sources.