Distances

Determination of distances in space

For the determination of distances in the universe, there are different approaches, which differ in whether the expansion of the universe (space expansion) is included in the consideration, and for which point of the time ray of the universe the distance is to be calculated.

In astronomy the following distances are calculated:

• Luminosity distance D_Lum taking into account the expansion of space and the associated redshift

This is based on a comparison of differences in brightness between two points in time that occur with the determination of the comoving distance D_com. a is a scale factor, which describes the space expansion in dependence of the redshift z. (a = 1/(1+z))

If the present time is used for z1, this results in the formula:

Or according to Etherington's reciprocity theorem:

Alternatively, it also applies:

Where F is the power density in [W/m²] and L is the luminosity in [W].

• naive Hubble D_H

A linear relationship postulated by Hubble and Lemaitre between the redshift z due to the expansions of space (escape velocity) and the distance. H₀ is the Hubble parameter that depends on the space extent.

• Comoving distance (proper distance at the present time) D_com

It is the "real" distance measured at the present time. It considers the space expansion occurring since the big bang and changes with every second. The comoving distance is always larger than the light travel time, because the light needs time to reach us or cannot reach us at all from a certain time. Expressed in the comoving distance, the universe has expanded since the big bang by 46.6 Gly.

• Light travel time distance D_T

It is the distance the photons have traveled since they left the object. It is often used in popular science publications as an indication of distance. Expressed in light travel time, the universe has expanded by 13.8 Gly. since the Big Bang.

• Angular diameter distance D_ang

This uses the extent of the object and geometric formulas to determine the distance. It is the distance that existed when the light left the object and thus does not reflect the actual distance.

To clarify it once again: An object, which emitted light, which reaches us now, shortly after the big bang, is 13.8 Gly. away from us after the light running time distance. Since the space has expanded during many times faster than the light, the object is 46.6 Gly. away from us. This also means that we will no longer see events that such an object emits at the present time, since the light will no longer reach us due to the faster-than-light expansion of space.

Each of these distances are calculated by different formulas, in which the space expansion is more or less included. The formulas can be viewed at https://en.wikipedia.org/wiki/Distance_measure.

As a consequence, the results vary as a function of the object distance and they are thus dependent on the resulting redshift z, due to the space expansion.

If the results about the redshift and the resulting distances of the mentioned formulas are shown in a graph, the following diagram results:

Source of left image: Wesino at English Wikipedia, Public domain, via Wikimedia Commons
Source of right image: Wesino at English Wikipedia, Public domain, via Wikimedia Commons

For small redshifts z all formulas agree, for more and more larger values the results drift apart.

If a flat (Ω₀ + Ω_Λ = 1) and matter-dominated (Ω_R ≈ 0) universe is assumed, the following, currently valid in science, values (measurements from the Planck satellite 2018) can be used for the formulas:

• Matter density parameter Ω₀ (dark matter (0.266) + baryonic matter (0.049)) = 0.315
• Vacuum energy density parameter Ω_Λ (dark energy) = 0.685
• Hubble parameter H₀ = 67.4 km/(s*Mpc)

There are two other possibilities to determine distances in space. These two possibilities refer to objects which are inside the Milky Way or near it and thus are not subject to the expansion of space, but highest to the Doppler effect (red or blue shift due to the change of distance of the object).

The first possibility refers again to the luminosity. But this time the formula does not depend on the redshift, but on the apparent magnitude m related to the absolute magnitude M of an object. (https://en.wikipedia.org/wiki/Luminosity_distance)

A second possibility is the determination of the distance via the parallax angle. Here, the apparent displacement of the position of a nearby object is considered as the position of the Earth changes in its orbit around the Sun.

Image source: Kes47 / Original version from German Wikipedia. By user: WikiStefan. 28 Oct 2004, CC BY 3.0, via Wikimedia Commons (supplemented with descriptions)

The formula is mainly used for the distance of stars within the Milky Way.

Due to the latest Gaia satellite data, somewhat further distance determinations outside the Milky Way are now possible. (http://spiff.rit.edu/classes/ladder/lectures/parallax/parallax.html, Section: How far can it reach? (Gaia))

For this purpose, a star name can be entered on the Gaia page (https://gea.esac.esa.int/archive/) via the search, and the parallax angle (in milliarcseconds) is then output.