concept

photometric distance

Luminosity distance is the distance inferred from the observed flux of a standard candle.
15 related articles · 6 formulas · 3 who worked here

Description

Photometric distance D_L relates the observed radiation flux F to the true luminosity L via the formula F = L / (4π D_L²). It includes a correction for the expansion of the universe: photons lose energy due to redshift (wavelength increases), and the time between the arrival of successive photons also stretches. As a result, D_L = (1+z) D_M, where z is the redshift, and D_M is the comoving transverse distance. For nearby objects, D_L ≈ D_M, but for distant galaxies (z ~ 1 and higher) the difference becomes significant. D_L is precisely what is measured using standard candles—objects of known luminosity, like Type Ia supernovae.

History

The concept originates from the work of Edwin Hubble, who in 1929 established the proportionality of redshift to distance. The formula for photometric distance in relativistic cosmology was derived within the framework of the Friedmann–Lemaître–Robertson–Walker metric. The Supernova Cosmology Project (Saul Perlmutter) and the High-Z Supernova Search Team (Brian Schmidt, Adam Riess) in 1998 used Type Ia supernovae as standard candles and discovered a deviation of D_L from the linear law, which indicated accelerated expansion. This became direct observational evidence for the existence of dark energy. Adam Riess and Saul Perlmutter are present in the list of scientists.

How it works

The relationship between apparent magnitude m and absolute magnitude M is given by the distance modulus: m − M = 5 log₁₀(D_L / 10 pc). For an object with known absolute luminosity (standard candle), measuring the peak apparent magnitude yields D_L. Within general relativity, D_L is expressed through solutions to the Friedmann equations, which follow from Einstein's equations for a homogeneous isotropic universe. The value of D_L depends on the densities of matter Ω_m, dark energy Ω_Λ, curvature Ω_k, and the Hubble constant H₀. Therefore, measuring D_L for a set of objects at different z allows placing constraints on cosmological parameters, in particular on the equation of state of dark energy w.

In practice

The method is used to measure distances to Type Ia supernovae, which led to the discovery of the accelerating expansion of the universe. Without it, modern cosmology would lose its ruler.

Fun fact

Looking at a galaxy with redshift z=1, we receive light dimmed by a factor of four just due to the expansion of space—the photometric distance turns out to be twice the comoving distance.

Concept graph →

Formulas

$$d = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$$
Euclidean distance measures straight-line distance between two points.
What is in the formula
dvariableEuclidean distance — distance metre
x₁variablex-coordinate of point 1 — coordinate metre
x₀variablex-coordinate of point 0 — coordinate metre
y₁variabley-coordinate of point 1 — coordinate metre
y₀variabley-coordinate of point 0 — coordinate metre
√operatortakes the square root of the sum — square root
d_i=\\sqrt{(\{pi}_i-\{pi}_0)^2+(\{le}_i-\{le}_0)^2} · 2606.01855v1
$$d_{ls} = d_s \sqrt{1 + \Omega_K d_l^2} - d_l \sqrt{1 + \Omega_K d_s^2}$$
Lens-source distance depends on curvature and observer distances.
What is in the formula
d_{ls}variableangular diameter distance between lens and source — angular diameter distance metre
d_svariableangular diameter distance from observer to source — angular diameter distance metre
d_lvariableangular diameter distance from observer to lens — angular diameter distance metre
\Omega_Kvariablecurvature density parameter — curvature density parameter
\frac{d_{ls}}{d_s}=\sqrt{1+ω_kd_l^2}-\frac{d_l}{d_s}\sqrt{1+ω_kd_s^2} · 2606.03634v1
d_{δt}=(1+z_l)\frac{d_ld_s}{d_{ls}} · 2606.25205
$$D_L(z) = \frac{c(1+z)}{H_0} \times \begin{cases} \frac{\sinh\left(\sqrt{\Omega_k}\chi(z)\right)}{\sqrt{\Omega_k}} & \Omega_k>0 \\ \chi(z) & \Omega_k=0 \\ \frac{\sin\left(\sqrt{-\Omega_k}\chi(z)\right)}{\sqrt{-\Omega_k}} & \Omega_k<0 \end{cases}$$
Luminosity distance depends on redshift and cosmological parameters.
What is in the formula
D_Lvariableluminosity distance — photometric distance metre
zvariableredshift — redshift
χvariablecomoving distance — comoving distance metre
Ω_kvariablecurvature density parameter — curvature density parameter
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
H_0constantcurrent expansion rate of the universe — hubble constant = 2.2e-18 per second
sinhoperatorhyperbolic sine function — hyperbolic sine
sinoperatorsine function — sine
√operatorsquare root — square root
d_l(z)=\frac{c(1+z)}{h_0}*\begin{cases}\frac{\sinh(\sqrt{ω_k}χ(z))}{\sqrt{ω_k}}&ω_k>0\χ(z)&ω_k=0\\frac{\sin(\sqrt{-ω_k}χ(z))}{\sqrt{-ω_k}}&ω_k<0\end{cases},\qua · 2606.04216v1
$$T_B = \frac{F_\nu c^2}{2 k_B \nu^2 \Omega}$$
Brightness temperature relates flux density to source temperature.
What is in the formula
T_Bvariablebrightness temperature — brightness temperature kelvin
F_νvariablespectral flux density — spectral flux density watt per metre squared per hertz
νvariablefrequency — frequency hertz
Ωvariablesolid angle of the source — solid angle steradian
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
k_Bconstantrelates energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin
t_b=\frac{f_{ν}c^2}{2k_bν^2ω}>10^{12}{k} · 2606.04232v1
t_b\sim\frac{\varepsilon_{\rmrad}r^2mnγ_jc^3}{2k_bν^3t^2} · 2606.25035
$$m - M = 5 \log_{10} \left( \frac{d}{10\,{\rm pc}} \right)$$
Relates apparent and absolute magnitudes to distance.
What is in the formula
mvariableapparent magnitude — apparent magnitude
Mvariableabsolute magnitude — absolute magnitude
dvariabledistance to the object — distance parsec
log10operatorbase-10 logarithm — logarithm base 10
m_{\rmbar}-m_{\rmbar}=5\log_{10}(\frac{d}{10{\rmpc}}) · 2606.05144v2
m-m=5\log_{10}(d/10) · 2606.26831
μ=m-m=5\log_{10}(d_l/{пк})-5 · 2607.03569
μ=m-m=5\log_{10}(\frac{d_l}{{мпк}})+25 · 2607.06959
$$m = H + 5\log_{10}\left(\frac{d}{10\,\text{pc}}\right) + \text{phase correction}$$
Apparent magnitude depends on absolute magnitude, distance, and phase angle.
What is in the formula
mvariableapparent magnitude — apparent magnitude
Hvariableabsolute magnitude — absolute magnitude
dvariabledistance to the object — distance parsec
\text{phase correctivariablecorrection for phase angle (e.g., for planets or asteroids) — phase correction
\log_{10}operatorlogarithm to base 10 — logarithm base 10
m=h+2.5\log_{10}(\frac{d_{os}^2d_{oe}^2}{q(α)d_{es}^4}) · 2606.05344v1

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