The HR Diagram

Section 6 of 6

Reference and Synthesis

Reference Tables

The Magnitude Scale: Key Values

Object	Apparent Magnitude $m$	Absolute Magnitude $M$
Sun	$-26.74$	$+4.83$
Full Moon	$-12.7$	—
Venus (brightest)	$-4.6$	—
Sirius (brightest star)	$-1.46$	$+1.42$
Vega	$+0.03$	$+0.58$
Naked-eye limit	${\sim}+6$	—
Hubble Space Telescope limit	${\sim}+31$	—

Main-Sequence Properties by Spectral Type

Spectral Type	$T_{\text{eff}}$ (K)	Mass ( $M_\odot$ )	Radius ( $R_\odot$ )	$L$ ( $L_\odot$ )	$M_V$	Main-Seq Lifetime
O5	$4.2 \times 10^4$	$40$	$12$	$5 \times 10^5$	$-5.7$	${\sim}4.5~\text{Myr}$
B0	$3 \times 10^4$	$15$	$7$	$3 \times 10^4$	$-4.0$	${\sim}10~\text{Myr}$
A0	$9{,}900$	$2.5$	$2.5$	$40$	$+0.6$	${\sim}1~\text{Gyr}$
F0	$7{,}200$	$1.6$	$1.5$	$6.5$	$+2.7$	${\sim}3~\text{Gyr}$
G2 (Sun)	$5{,}800$	$1.0$	$1.0$	$1.0$	$+4.8$	${\sim}10~\text{Gyr}$
K0	$5{,}300$	$0.8$	$0.85$	$0.4$	$+5.9$	${\sim}20~\text{Gyr}$
M0	$3{,}850$	$0.5$	$0.6$	$0.08$	$+8.8$	${\sim}60~\text{Gyr}$
M5	$3{,}000$	$0.1$	$0.15$	$0.001$	$+12.3$	over $100~\text{Gyr}$

Order-of-magnitude only; values depend on metallicity, rotation, and mass loss.

Summary: Finding Patterns, Needing Models

The magnitude system is a logarithmic brightness scale where 5 magnitudes = a factor of 100 in flux. Absolute magnitude removes distance, placing all stars at a standard $10~\text{pc}$ .
The distance modulus ( $m - M = 5\log_{10}(d/10\,\text{pc})$ ) connects observed brightness, intrinsic brightness, and distance — the inverse-square law in logarithmic form.
The observer’s HR diagram ( $M_V$ vs. spectral type) is pure measurement, revealing the main sequence, giant branch, and white dwarf sequence.
The theorist’s HR diagram ( $\log L$ vs. $\log T_{\text{eff}}$ ) overlays lines of constant radius from Stefan-Boltzmann — giants are enormous, white dwarfs tiny.
The main sequence is a mass sequence — ${\sim}0.08\,M_\odot$ (lower right) to ${\sim}150\,M_\odot$ (upper left). The mass-luminosity relation made visible.
The HR diagram is an evolution diagram. Stars move as they age; mass determines the path. The main sequence is where stars live; the giant branch where they age; the white dwarf sequence where they end up.
Patterns demand physics. Why does the main sequence exist? Why do stars become giants? What sets the maximum white dwarf mass? Module 3 answers these.

Observable → Model → Inference: The HR Diagram Chain

Observable

Apparent brightness and color of many stars, plus parallax distances

Photometry and spectroscopy give apparent magnitude and spectral type (color); parallax gives distance — together yielding absolute magnitudes and color indices for each star.

Model

The magnitude / distance-modulus / Stefan-Boltzmann calibration stack

The magnitude system converts flux ratios to a log scale; the distance modulus (inverse-square law) connects apparent and absolute magnitude; Wien’s law connects color to temperature; Stefan-Boltzmann connects $L$ and $T$ to radius; structure models connect $L$ and $T$ to mass and evolutionary state.

Inference

HR structure — a mass-ordered main sequence, giants, and white dwarfs

Stars cluster into a main sequence (a mass sequence), a giant branch (evolved, bloated envelopes), and a white dwarf sequence (dead cores). Mass, on neither axis, organizes the whole pattern — and the patterns become the questions Module 3 must answer.

Glossary

Absolute magnitude: The apparent magnitude a star would have if placed at the standard distance of $10~\text{pc}$ ( $M$ ). It expresses intrinsic luminosity in the magnitude system, removing the distance dependence.
Apparent magnitude: A logarithmic measure of how bright a star appears from Earth ( $m$ ), set by both luminosity and distance. Smaller (more negative) means brighter; 5 magnitudes correspond to a factor of 100 in flux.
Color-magnitude diagram: The observer’s form of the HR diagram: absolute magnitude versus color index (or spectral type), built from photometry and distance alone — no physical theory required.
Distance modulus: The difference $m - M = 5\log_{10}(d/10\,\text{pc})$ between apparent and absolute magnitude — a logarithmic readout of distance. Zero at $10~\text{pc}$ ; the inverse-square law in magnitude form.
Hertzsprung-Russell diagram: A plot of stellar luminosity (or absolute magnitude) against effective temperature (or spectral type) that organizes stars into the main sequence, giant branch, and white dwarf sequence. The single most important diagram in astrophysics — a classification chart, a mass map, and an evolution diagram at once.
Luminosity class: The Roman-numeral part of a stellar classification (I supergiant, III giant, V dwarf, …), read from spectral line widths (surface gravity). It distinguishes stars of equal temperature but very different size and luminosity.
Main sequence: The diagonal band on the HR diagram where hydrogen-core-burning stars spend ~90% of their lives, ordered by mass: high-mass stars are hot and luminous (upper left), low-mass stars cool and faint (lower right). It is the mass-luminosity relation made visible.
Red giant: An evolved star that has exhausted core hydrogen and expanded enormously (typically $10\text{–}100\,R_\odot$ ), becoming cool but very luminous — the upper-right region of the HR diagram.
Spectroscopic parallax: A distance method (despite the name, unrelated to parallax): a main-sequence star’s spectral type fixes its absolute magnitude, and the distance modulus then yields the distance from the apparent magnitude.
Supergiant: The most luminous evolved stars, with radii reaching $100\text{–}1{,}000\,R_\odot$ (luminosity class I) — e.g. Betelgeuse. They occupy the very top of the HR diagram across a wide temperature range.
White dwarf: The Earth-sized ( $R \sim 0.01\,R_\odot$ ) remnant core of a low- or intermediate-mass star — hot but faint, in the lower-left of the HR diagram, cooling slowly with no ongoing fusion.