Vol. Science · Astronomy · Standard candles

Distance ladder
one rung at a time

A cosmic distance is a chain of overlapping measurements. The shared equation is the distance modulus, m − M = 5 log₁₀(d / 10 pc), with an optional extinction A in the same band. At 10 pc and A = 0, m = M. The first rung is parallax, d(pc) = 1 / p(″), the same reciprocal as the parallax desk, plus the way a fractional parallax error becomes a distance error. Cepheids use Henrietta Leavitt’s period–luminosity relation: a period fixes an absolute magnitude, and an apparent mean magnitude then fixes a distance. Type Ia supernovae are the brighter candle. At low redshift, d ≈ cz / H₀, and the two cited Hubble constants do not give the same distance. The wavelength shift itself is the redshift desk. Age, lookback, and the distances past the cz/H₀ toy are the cosmology desk. Brightness without the candles is magnitude. Luminosity on the HR diagram is not a distance.

Educational approximation. Not a catalog and not a cosmological fit. Not a stellar-evolution model. The bars are a sketch of rough ranges, and they overlap on purpose. Extinction is optional and has to be the same band as the magnitudes. The Cepheid line is a mean of ten Galactic stars. The Type Ia number is a mean of many events, not the magnitude of one supernova. cz/H₀ is the low-redshift toy, and peculiar velocities matter. The Friedmann integral is the cosmology desk. The Lutz–Kelker–Hanson correction is not applied. Real preset numbers are the cited ones. A typed number is arithmetic, not a new object.

Interactive · Modulus, parallax, Cepheid, Type Ia, Hubble flow

Log distance · approximate ranges · click a bar

Parallax
Cepheid
Type Ia
Hubble flow
This distance

Sketch, not a catalog limit. Parallax is drawn from Proxima’s 1.3 pc out to 100 pc, a stand-in for “a few hundred light-years.” NASA does not print 100 pc.

Rung

Presets · cited objects only

Solve for

m mag

M mag

A mag

Distance unit · input when distance is known

The sliders need JavaScript. The anchors are still these. m = 10, M = 0, A = 0 gives d = 1000 pc exactly. A = 1 gives d = 10 × 10^1.8 pc. The Sun at 10 pc stays at +4.83, and 10 pc = 32.616 ly. Sirius: ESA prints p = 0.37921″ and 1/0.37921 = 2.637 pc. No uncertainty is printed there. δ Cephei parallax: 3.66 ± 0.15 mas, so d = 1000/3.66 pc, the fraction is 0.15/3.66, the near bound is 1000/3.81 pc, and the far bound is 1000/3.51 pc. Benedict et al. 2007, Table 12 column B07, Johnson V: M_V = −4.05 ± 0.02 + (−2.43 ± 0.12)(log₁₀ P − 1). P = 10 days gives M_V = −4.05. δ Cephei in that paper’s Table 2: P = 5.36627 d, ⟨V⟩ = 3.960, A_V = 0.23. Type Ia: Richardson et al. 2014 bias-corrected normal mean M_B = −19.25 ± 0.20. SN 2011fe: B max = 10.00 ± 0.02 and A_B = 0.11 ± 0.03, so μ = 29.14 and d = 10 × 10^29.14/5 pc. Richmond and Smith adopt (m−M) = 29.10 ± 0.15 for M101, a different number. Coma: Colless and Dunn 1996, mean cz = 6853 km s⁻¹. Planck H0 = 67.4 gives 6853/67.4 Mpc. SH0ES H0 = 73.04 gives 6853/73.04 Mpc.

One candle is not the ladder

The mean line is not the star, and cz is not a ruler past the neighborhood.

The Cepheid calculator uses Benedict et al. 2007, Table 12, column B07, with the Lutz–Kelker–Hanson correction already inside the published fit: M_V = −4.05 ± 0.02 + (−2.43 ± 0.12)(log₁₀ P − 1), Johnson V. That is a straight line through ten Galactic Cepheids. The zero-point is at log P = 1, so a 10-day period gives M_V = −4.05 exactly. Their column B07f, without LKH, is −4.03 and −2.44. This page does not switch to that column. The ±0.02 and ±0.12 are the paper’s coefficient uncertainties. They are not folded into a distance error here, because the covariance is not copied. Section 6.1 gives the standard deviation of the V residuals as 0.10 mag. A shift of 0.10 mag moves distance by a factor 10^0.02. That factor is an illustration of the scatter. It is not an error bar on one star.

δ Cephei’s own parallax absolute magnitude in Table 11 is M_V = −3.47 ± 0.11, with (m−M)₀ = 7.19 and an LKH correction of −0.01 mag. The plain inversion d = 1000/3.66 pc is not that modulus. This desk does not apply LKH on top of 1/p. The Cepheid preset compares the mean line with that Table 11 magnitude. They are allowed to disagree. The star is one point. The line is the mean.

The Type Ia candle is Richardson, Jenkins, Wright, and Maddox 2014, Table 1, bias-corrected normal supernovae: M_B = −19.25 ± 0.20, standard deviation 0.50 mag, N = 239. The ±0.20 is the uncertainty on the mean. The 0.50 mag is the width of the distribution. One event is not known to 0.20 mag. No decline-rate correction is applied. Hillebrandt and Niemeyer 2000 quote M_B ≈ M_V ≈ −19.30 ± 0.03 + 5 log(H₀/60), with a dispersion of 0.3 mag or less. That −19.30 is the value at H₀ = 60. It moves when H₀ moves. The calculator holds Richardson’s −19.25.

Hubble flow on this page is d = cz/H₀ with the defined speed of light, the same non-relativistic product as the redshift desk. Hogg’s notes limit that linear law to small redshift. Coma’s mean cz is 6853 km s⁻¹, so z = 6853/c is still about 0.023. The velocity distribution is non-Gaussian, and the cluster is a merger. The dispersion σ_cz = 1082 km s⁻¹, divided by either H₀, is a larger shift than the gap between Planck and SH0ES. The smaller Hubble constant is the larger distance. Neither line is a new measurement of Coma.

Literacy · the numbers on the readout

Distance modulus: m − M = 5 log₁₀(d / 10 pc) + A. ESA’s teacher notes write M = m − 5 log(D/10), with D in parsecs, and no extinction. A = 0 is that formula, and it is the magnitude desk. A is extinction in the same band, subtracted before the modulus the way Benedict subtracts A_V from ⟨V⟩. At 10 pc and A = 0, m = M. The Sun’s absolute visual magnitude on the NASA Sun Fact Sheet is +4.83. Light-years use 3.2616 per parsec, the magnitude desk’s four-decimal rounding. 10 pc is 32.616 ly.
Parallax: d(pc) = 1/p(″). The fractional error σ_d/d equals σ_p/p. The inverted bounds 1/(p±σ) are not symmetric: the far side is larger. If σ ≥ p the far bound is unbounded. Sirius on ESA’s stellar-distances page is p = 0.37921″, and they print 1/0.37921 = 2.637 pc. That page gives no uncertainty, so this one does not invent one.
δ Cephei parallax: Benedict et al. 2007, Table 11, repeating Benedict et al. 2002: π_abs = 3.66 ± 0.15 mas. Plain inversion is 1000/3.66 pc. The fraction is 0.15/3.66. The near bound is 1000/3.81 pc and the far bound is 1000/3.51 pc. Table 11 also prints M_V = −3.47 ± 0.11, (m−M)₀ = 7.19, and an LKH correction of −0.01 mag. The plain modulus is not 7.19.
Cepheid line: Benedict et al. 2007, AJ 133, 1810, Table 12, column B07, Johnson V, absorption and LKH corrected: M_V = −4.05 ± 0.02 + (−2.43 ± 0.12)(log₁₀ P − 1). The ten-star window used here is FF Aql at log P = 0.6504 (section 6.1) through ℓ Car at log P = 1.5509 (Table 2). Outside that window the calculator still runs and says so. δ Cephei’s Table 2 row is P = 5.36627 d, printed log P = 0.7297, ⟨V⟩ = 3.960, A_V = 0.23. The extinction-corrected mean magnitude is 3.730. The distance uses log₁₀ of 5.36627, not a second slope.
Type Ia mean: Richardson et al. 2014, Table 1, bias-corrected normal Ia, B band: M_B = −19.25 ± 0.20, standard deviation 0.50 mag, N = 239. The volume-limited mean in that table is −19.26 ± 0.20. The fixed candle is −19.25. Hillebrandt and Niemeyer’s −19.30 is the H₀ = 60 figure, not this candle.
SN 2011fe: Richmond and Smith 2012, Table 5: B max = 10.00 ± 0.02, V max = 9.99 ± 0.01. They compute A_B = 0.11 ± 0.03 and A_V = 0.08 ± 0.02 from Schlegel, Finkbeiner, and Davis 1998, with the uncertainty doubled. This desk uses B and A_B with M_B = −19.25, so μ = 29.14. They adopt (m−M) = 29.10 ± 0.15 for M101 as a rough average of other work. That adopted modulus is shown beside the candle. The two are not forced to agree. The host is M101.
Hubble flow: d ≈ cz/H₀, with c = 299792.458 km/s. Planck Collaboration 2020, base-ΛCDM, H₀ = 67.4 ± 0.5. Riess et al. 2022, SH0ES, H₀ = 73.04 ± 1.04. Both central values are drawn. The marker follows the one you select. Colless and Dunn 1996: the main Coma cluster, centered on NGC 4874, has mean cz = 6853 km s⁻¹ and σ_cz = 1082 km s⁻¹. The smaller H₀ is the larger distance. The dispersion scale is larger than that H₀ gap. The full redshift treatment, including why cz is not a speed at large z, is the redshift desk.
Ladder sketch: Approximate ranges, not catalog limits. Parallax: 1.3 pc, the NASA Hubble caption for Proxima, out to 100 pc. 100 × 3.2616 = 326.16 ly, a sketch of NASA’s “a few hundred light-years.” NASA does not print 100 pc. Cepheid: from the δ Cephei plain inversion, which is a point on the rung and not “where Cepheids begin,” out to ∼40 Mpc. Type Ia: from the distance of Richmond and Smith’s adopted M101 modulus 29.10, a sketch near that galaxy, out to 600 Mpc. Hubble flow: ∼100 to 600 Mpc. The tildes on 40, 100, and 600 Mpc are Scolnic and Vincenzi’s. The bars overlap.

Sources · the coefficients and the presets

Cited, not invented

ESA Science & Technology — Teacher’s notes, stellar distances — M = m − 5 log(D/10), D in parsecs. A = 0 on this desk is that relation. The magnitude desk uses it with extinction left out.
NASA NSSDCA — Sun Fact Sheet — absolute magnitude +4.83. The 10 pc button. Apparent magnitude at that distance is the same number.
IAU 2012 Resolution B2 — the parsec is (648000/π) AU. The light-year count here is 3.2616, the four-decimal rounding used on the magnitude desk. NASA and ESA round the same length to 3.26 ly.
ESA Science & Technology — Stellar distances — d(parsec) = 1 / p(arcsecond). Sirius is the worked example: parallax 0.37921 arcseconds, distance 1/0.37921 = 2.637 parsecs. No uncertainty is printed on that page.
NASA Science — Hubble image of ESO 300-16 — Proxima Centauri at 1.3 parsecs. The left edge of the parallax bar. One parsec is 3.26 light-years on that caption.
NASA Science — Hubble stretches the stellar tape measure — parallax works for stars within a few hundred light-years. The right edge of the parallax bar is drawn at 100 pc, which is 326.16 ly on this desk’s light-year. That 100 pc is a sketch of the sentence. NASA does not print 100 pc.
Benedict et al. 2007, AJ 133, 1810 (arXiv:astro-ph/0612465) — Table 12, column B07, Johnson V: M_V = −4.05 ± 0.02 + (−2.43 ± 0.12)(log₁₀ P − 1), with LKH. Column B07f, without LKH, is −4.03 and −2.44. Table 2: δ Cephei P = 5.36627 d, log P = 0.7297, ⟨V⟩ = 3.960, A_V = 0.23; ℓ Car log P = 1.5509. Table 11: δ Cephei π_abs = 3.66 ± 0.15 mas, M_V = −3.47 ± 0.11, (m−M)₀ = 7.19, LKH correction −0.01 mag. Section 6.1: FF Aql log P = 0.6504, and the standard deviation of the V residuals is 0.10 mag. The parallax row repeats Benedict et al. 2002, AJ 124, 1695.
Richardson, Jenkins, Wright, and Maddox 2014, AJ 147, 118 (arXiv:1403.5755) — Table 1, bias-corrected normal Type Ia: M_B = −19.25 ± 0.20, standard deviation 0.50 mag, N = 239. Volume-limited M_B = −19.26 ± 0.20. The candle is −19.25.
Hillebrandt and Niemeyer 2000, ARA&A 38, 191 — eq. (1): M_B ≈ M_V ≈ −19.30 ± 0.03 + 5 log(H₀/60), dispersion σ_M ≤ 0.3, citing Hamuy et al. 1996b. The −19.30 is the value at H₀ = 60. It is not the candle on the calculator.
Richmond and Smith 2012, arXiv:1203.4013 — SN 2011fe in M101. Table 5: B max = 10.00 ± 0.02, V max = 9.99 ± 0.01. Extinction from Schlegel, Finkbeiner, and Davis 1998, uncertainty doubled: A_B = 0.11 ± 0.03, A_V = 0.08 ± 0.02. They adopt (m−M)_M101 = 29.10 ± 0.15 as a rough average of other work. Δm₁₅(B) = 1.21 ± 0.03. This desk does not apply a decline-rate correction.
Colless and Dunn 1996, ApJ 458, 435 (arXiv:astro-ph/9508070) — abstract: the main Coma cluster, centered on NGC 4874, mean cz = 6853 km s⁻¹, σ_cz = 1082 km s⁻¹. A separate NGC 4839 group is not the preset. The velocity distribution is non-Gaussian. This page does not assign a velocity frame the paper’s abstract does not state.
Planck Collaboration 2020, A&A 641, A6 (arXiv:1807.06209) — abstract, base-ΛCDM: H₀ = (67.4 ± 0.5) km s⁻¹ Mpc⁻¹. The central value is the Planck line.
Riess et al. 2022, ApJL 934, L7 (arXiv:2112.04510) — baseline SH0ES H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹. The central value is the SH0ES line.
Hogg 1999, arXiv:astro-ph/9905116 — the linear Hubble law holds at small redshift. The product cz/H₀ on this page is that toy. The shift, the relativistic Doppler speed, and the scale-factor statement are the redshift desk.
Scolnic and Vincenzi, arXiv:2311.16830 — HST Cepheids limited to ∼40 Mpc. Hubble flow ∼100 to 600 Mpc. The tildes are theirs. The Type Ia bar is drawn out to that 600 Mpc figure, and the Hubble-flow bar uses both ends. Those edges are the sketch, not a new limit.

Classroom distance ladder. m − M = 5 log10(d / 10 pc) + A. Parallax d(pc) = 1/p(″), with the asymmetric bounds 1/(p±σ). Cepheid M_V from Benedict et al. 2007, Table 12, column B07. Type Ia M_B = −19.25 from Richardson et al. 2014. Hubble flow d = cz/H0 with Planck 67.4 and SH0ES 73.04. Presets use the cited central values only. The bars are approximate ranges. Not a catalog. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

First rung

Axis	Log distance, from 1 pc to 1 Gpc. Ticks at 1 pc, 1 kpc, 1 Mpc, and 1 Gpc.
Bars	Parallax, Cepheid, Type Ia, Hubble flow. Click a bar to select that rung. The ranges are a sketch.
Marker	The distance the current rung computes. On Hubble flow, a second tick shows the other H0.

Diagram	The canvas takes focus. Left and Right change the rung. Click or tap a bar to select it.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Distance ladder
one rung at a time

The mean line is not the star, and cz is not a ruler past the neighborhood.

Cited, not invented

Parallax

Magnitude

Redshift

Cosmology

HR diagram

Nearest stars

Galaxy

Distance ladder

The diagram

Controls

Keyboard and touch

Modulus	Solve for distance, apparent m, or absolute M. A is optional extinction in the same band. Empty A is zero. Distance can be read in pc, ly, or Mpc.
Parallax	d = 1/p. Switch mas or arcsec. Leave σ empty when the source prints none. The far bound is unbounded when σ ≥ p.
Cepheid	Period from 2 to 60 days. M_V comes from the Benedict B07 line. Distance uses ⟨V⟩ and A_V. Outside the ten-star window, the card says so.
Type Ia	Peak apparent B and A_B. M_B stays at −19.25.
Hubble flow	cz from 200 to 30,000 km/s. Both cited H0 distances stay on the card. The marker follows Planck or SH0ES.
Presets	Sun at 10 pc, Sirius, δ Cephei parallax, δ Cephei as a Cepheid, SN 2011fe, and the Coma cluster. Editing a field leaves the preset. Switching H0 on Coma keeps it.

Distance ladderone rung at a time

The mean line is not the star, and cz is not a ruler past the neighborhood.

Cited, not invented

Parallax

Magnitude

Redshift

Cosmology

HR diagram

Nearest stars

Galaxy

Distance ladder
one rung at a time