Distance & Parallax

Section 7 of 7

Standard Candles and Synthesis

Part 7: Standard Candles — A Preview

Parallax reaches ~100 pc from the ground, ~10 kpc with Gaia — too close for most of the Milky Way, let alone other galaxies. Beyond that, we invert the inverse-square law: if we know a source’s intrinsic luminosity, we measure its flux and infer distance, $d = \sqrt{L/(4\pi F)}$ . Such objects of known luminosity are standard candles (defined in Lecture 1).

Worked Example 4Standard-Candle Distance

Problem

A Cepheid has known $L = 10{,}000\,L_\odot = 3.8 \times 10^{37}\ \mathrm{erg\,s^{-1}}$ (from its pulsation period) and measured $F = 1.0 \times 10^{-14}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ . Find its distance.

StepInvert the inverse-square law

$d = \sqrt{\frac{L}{4\pi F}} = \sqrt{\frac{3.8\times10^{37}}{4\pi (1.0\times10^{-14})}} \approx \sqrt{3.0\times10^{50}\ \text{cm}^2} \approx 1.7 \times 10^{25}\ \text{cm}$

Dimensional check

$\sqrt{\mathrm{erg\,s^{-1}} / (\mathrm{erg\,cm^{-2}\,s^{-1}})} = \sqrt{\text{cm}^2} = \text{cm}$ ✓.

Result

$d \approx 1.7 \times 10^{25}$ cm $\approx 5.5$ Mpc — well beyond the Milky Way, in a nearby galaxy, far past parallax reach.

Two-panel diagram: Left shows Cepheid Variable with sinusoidal light curve where pulse rate reveals wattage. Right shows Type Ia Supernova with white dwarf accreting from companion star, producing consistent peak brightness (~10^9 solar luminosities). Bottom equation: Measure Flux (F) + Know Luminosity (L) -> Calculate Distance (d). — Figure 10Standard candles work because physics predicts their luminosity. Measure flux (F) + know luminosity (L) to calculate distance (d).Course illustration (A. Rosen)

Cepheid variables pulsate over days to months; Henrietta Leavitt (1912) found that pulsation period tracks luminosity, so measuring the period gives $L$ . Hubble used Cepheids in Andromeda in the 1920s to prove it is a separate galaxy. Type Ia supernovae — thermonuclear explosions of white dwarfs — reach a consistent peak ( $\sim$ a few $\times 10^9\,L_\odot$ ), making them standard candles to cosmological distances; in 1998 they revealed the accelerating expansion (dark energy, 2011 Nobel Prize).

Four-rung ladder diagram titled 'The Cosmic Distance Ladder: Building on the Shoulders of Physics'. From bottom to top: Rung 1 Geometry (Parallax), Rung 2 Physics (Cepheids/Standard Candles), Rung 3 Physics (Supernovae/Chandrasekhar Limit), Rung 4 Cosmology (Hubble Flow). Footer text: Our understanding of the vastest scales relies on the microscopic atom. — Figure 11Each rung calibrates the next. Parallax (geometry) to Cepheids (standard candles) to Supernovae (Chandrasekhar limit) to Hubble Flow (cosmology). We infer the infinite from the infinitesimal.Course illustration (A. Rosen)

The cosmic distance ladder chains the methods: parallax (geometry) calibrates nearby Cepheids; Cepheids calibrate Type Ia supernovae; each rung rests on the one below. The entire cosmic distance scale ultimately rests on getting nearby parallaxes right — which is why Gaia’s improvement rippled through all of cosmology.

Summary: Observable → Model → Inference

Goal	Observable	Model	Inference
Distance	Position shift across the sky (arcsec)	Parallax: $d(\text{pc}) = 1/p('')$	Distance in pc and cm
Luminosity	Measured flux (photometry)	Inverse-square law $F = L/4\pi d^2$	$L = 4\pi d^2 F$
Distance (reversed)	Flux + known luminosity (Cepheid period)	Standard-candle relation	$d = \sqrt{L/4\pi F}$

Each step has assumptions: parallax assumes Earth’s orbit is known; the inverse-square law assumes isotropic emission and no absorption; standard candles assume correct calibration. Good science always asks: what could go wrong?

Looking Ahead

This lecture gave you distance and luminosity — one axis of the HR diagram. Next (Surface Flux & Colors) we apply thermal physics (Stefan-Boltzmann, Wien) to get effective temperature and radius; later lectures add composition (spectra), mass (binary orbits), and the magnitude system, before we assemble the full HR diagram.

Glossary

Arcsecond: An angular unit equal to $1/3600$ of a degree (or $1/60$ of an arcminute) — the standard unit for stellar parallax. It measures angle, never distance.
Hertzsprung-Russell diagram: A plot of stellar luminosity (vertical axis, logarithmic) versus effective temperature (horizontal axis) that reveals stellar properties and evolutionary states. Building it requires knowing luminosity — which requires distance.
Parallax angle: The apparent angular shift of a nearby star as Earth orbits, defined as half the total six-month shift — the angle Earth’s 1 AU orbital radius subtends as seen from the star. Measured in arcseconds.
Parsec: The distance at which a star shows a parallax of $1''$ (with Earth’s 1 AU baseline): $1\ \text{pc} = 206{,}265\ \text{AU} = 3.09 \times 10^{18}\ \text{cm} = 3.26$ light-years. The natural unit for stellar distances.
Radian: The natural (dimensionless) angle unit: the angle subtended by an arc whose length equals the radius, so angle = arc length / radius. A full circle is $2\pi$ rad, and $1\ \text{rad} = 206{,}265''$ .
Small-angle approximation: For small angles ( $\alpha \ll 1$ rad), $\sin\alpha \approx \tan\alpha \approx \alpha$ , giving the clean geometric relation $s = \alpha d$ between physical size, angular size, and distance.