Proper Distance Calculator Formula

Understand the math behind the proper distance calculator. Each variable explained with a worked example.

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Formulas Used

Proper Distance

proper_dist = scale_factor * comoving_distance

Proper Distance (Mly)

proper_dist_mly = scale_factor * comoving_distance * 3.2616

Variables

VariableDescriptionDefault
comoving_distanceComoving Distance(Mpc)428
scale_factorScale Factor a(t)1

How It Works

Proper Distance

The proper distance is the physical distance between two objects at a particular moment in cosmic time.

Relation

d_proper = a(t) * d_comoving

  • *a(t)* is the cosmic scale factor (a = 1 today)
  • *d_comoving* is the comoving distance

    • At the present epoch a = 1, so proper distance equals comoving distance. In the past (a < 1) the physical distance was smaller.

    Worked Example

    A galaxy with comoving distance 428 Mpc observed at the present epoch (a = 1).

    comoving_distance = 428scale_factor = 1
    1. 01d_proper = a(t) * d_comoving
    2. 02d_proper = 1.0 * 428
    3. 03d_proper = 428 Mpc ≈ 1396 Mly

    Frequently Asked Questions

    What is the scale factor?

    The scale factor a(t) describes how distances in the universe have changed over time. By convention a = 1 today; at the Big Bang a = 0.

    When would I use a scale factor other than 1?

    To find the physical distance at an earlier epoch. For instance, at redshift z the scale factor was a = 1 / (1 + z).

    Is the proper distance directly observable?

    Not directly. We infer it from redshift and a cosmological model. What we observe is the luminosity distance or angular-diameter distance.

    Ready to run the numbers?

    Open Proper Distance Calculator