iCosmos › Luminosity distance calculator
Luminosity distance calculator
The distance that converts an observed flux into a luminosity: F = L / (4π DL2).
The interactive calculator loads here: change the redshift and the cosmological parameters (Ωm, ΩΛ, Ωr, w0, wa, H0) and the result and plot update instantly.
What it is
The luminosity distance DL is defined so that the inverse-square law holds in an expanding universe: a source of luminosity L produces a flux F = L / (4π DL2). It is the distance measured with standard candles such as Type Ia supernovae, whose luminosity distances revealed in 1998 that the expansion of the universe is accelerating.
DL is larger than the transverse comoving distance DM (equal to DC in a flat universe) by a factor (1 + z). Each photon arrives with less energy (redshift) and the photons arrive less often (time dilation); each effect dims the flux by (1 + z), and a factor (1 + z)2 in flux is one factor (1 + z) in distance.
Formula
- DL = (1 + z) DM
- DL = (1 + z)2 DA
- μ = 5 log10(DL / 10 pc)
Worked example
In the classic model (Ωm = 0.3, ΩΛ = 0.7, H0 = 70 km s−1 Mpc−1), DL(z = 1) = 6,607.66 Mpc (21.55 billion light-years), giving a distance modulus μ = 44.10 mag.
At z = 0.5, typical of the supernovae used to discover cosmic acceleration, DL = 2,832.94 Mpc and μ = 42.26 mag.
Common questions
Why is the luminosity distance larger than the comoving distance?
Two effects dim a distant source by (1 + z) each: every photon is redshifted to lower energy, and photons arrive at a lower rate because of cosmological time dilation. Together they reduce the flux by (1 + z)2, which corresponds to one extra factor (1 + z) in distance: DL = (1 + z) DM, where DM equals the comoving distance DC in a flat universe.
How is luminosity distance used with supernovae?
Type Ia supernovae have nearly the same peak luminosity, so measuring their flux gives DL. Comparing DL(z) with the predictions of different cosmologies is how cosmic acceleration and dark energy were discovered.
Does this include a K-correction?
No. DL and μ here are purely cosmological (bolometric). Comparing magnitudes in a fixed filter also needs a K-correction for the source spectrum.
Related calculators
- Distance modulus calculator
- Angular diameter distance calculator
- Comoving distance calculator
- Redshift to distance calculator
- The full iCosmos cosmology calculator
About iCosmos
iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.
How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.