✨ Lecture 03 — Magnitudes, Colors & Spectral Classification

Summary

How to quantify stellar brightness — from Hipparchus’s scale to absolute magnitude — and how color indices derived from that scale let us infer temperature for thousands of stars without needing an individual spectrum for each one.

Lecture info

Course: Galactic Archaeology and Stellar Populations Institution: National Observatory (ON), Brazil Professor: Hélio Dotto Perottoni

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💡 Light propagation and the brightness–distance degeneracy

Luminous flux (energy emitted per unit area) decreases with the square of the distance to the source (inverse-square law). This creates a degeneracy between intrinsic brightness (luminosity) and distance: observing a bright object in the sky, apparent brightness alone can’t tell us whether it’s intrinsically luminous and far away, or faint and nearby. That’s why what we always observe directly is apparent brightness — measured in practice by counting the number of photons received (today, via CCD).

📏 The magnitude scale

Hipparchus (190–120 BC) established the first comparative scale of stellar brightness, from 1 (brightest) to 6 (limit of human vision). Between magnitudes 1 and 6 there’s a 100× difference in flux — so each magnitude step corresponds to a factor of $100^{1/5}\approx 2.512$ in flux. The formal definition:

$m_1-m_2=-2.5{\log }_{10}\left(\frac{F_1}{F_2}\right)$

The negative sign imposes the inverse relation between magnitude and brightness: the smaller the magnitude, the brighter the object.

Absolute magnitude

Absolute magnitude ($M$) is the magnitude a star would have if placed exactly 10 parsecs from the Sun — a measure of intrinsic brightness, free from the distance degeneracy. The difference between apparent and absolute magnitude is the distance modulus:

$m-M=5{\log }_{10}(d)-5(d\text{ in parsecs})$

This is one of the fundamental equations of Astronomy. Example: knowing that the distance modulus of the Large Magellanic Cloud (LMC) is 18.5, and that the Sun’s absolute magnitude is $\approx 4.8$, a solar-type star in the LMC would have apparent magnitude $m=18.5+4.8=23.3$.

🎨 Magnitude and photometric systems

• VEGA magnitudes: based on the star Vega, defined with $V\approx 0.03$ and colors $\approx 0$ by construction. The zero-point depends on Vega’s spectrum in each band.
• AB magnitudes: defined by a constant absolute physical flux (independent of any reference spectrum).
• griz / Gunn / Oke: based on observational calibration, historically tied to standard stars (e.g., F subdwarfs).

A magnitude system is not a filter system

You can use any filter within any magnitude system — the two are independent concepts. Several photometric systems have been developed for different applications and wavelength ranges [Almeida-Fernandes et al. 2021; Perottoni et al. 2024].

🌈 Color indices

In Astronomy, a color (or color index) is the difference between an object’s magnitude in two spectral bands — e.g., $B-V$ (Johnson/UBV system). In the absence of selective absorption, colors are independent of distance (the brightness degeneracy cancels out in the subtraction). Vega has all colors equal to 0 in the VEGAmag system, by construction.

Considering blackbody spectra for three stars with $T_a>T_b>T_c$:

• $T_a=30,000$K: flux in B greater than in V → $B-V<0$ (bluer)
• $T_b=10,000$K: flux in B $\approx$ flux in V → $B-V\sim 0$
• $T_c=3,000$K: flux in B smaller than in V → $B-V>0$ (redder)

Color indices are extremely useful in practice: they let you estimate a physical property (temperature) for thousands of stars at once, without the cost of obtaining an individual spectrum for each.

🔬 Spectral classification

Historical development

• Late 19th century (~1890): Harvard University obtains spectra for ~10,000 stars; Williamina Fleming develops the foundations of modern classification based on hydrogen line intensity; the Henry Draper Catalog (HD) is born.
• Early 20th century (~1910): with a sample of ~200,000 spectra, Annie Jump Cannon refines the classification by considering the correlation between spectral type and color (i.e., temperature) — the Harvard Classification is born.

The OBAFGKM sequence

Cannon’s classification uses 7 main classes, organized by decreasing temperature (not alphabetical order, since it’s an adaptation of Fleming’s original scheme):

Type	Temperature	Dominant lines
O	Hottest	He II (ionized)
B	Very hot	C, He I (neutral)
A	Hot	H (strongest in the entire sequence)
F–G	Intermediate	Metals in general (Sun is G)
K–M	Cool	Metal lines / molecules (TiO in M)

The peak of the class M spectrum is shifted to longer wavelengths, while type O emits most intensely at small $\lambda$ — Wien’s Law in action. Type A stars ($\sim$10,000 K) have the most intense hydrogen absorption lines of the entire sequence (see Lecture 04 for the physical explanation, via the population of hydrogen’s energy levels).

📷 Types of photometry

• Absolute photometry: measures brightness on a calibrated, physical scale, allowing comparison of objects across different sky regions (all sky). Requires a photometric night and calibration with standard stars — more sensitive to atmospheric variation.
• Differential photometry: measures brightness relative to other stars in the same field, observed simultaneously in the same image. Less affected by atmospheric conditions; works even without a perfectly photometric night.
• Time-domain photometry: tracks brightness variations of the same object over time (essential for identifying variables, such as the Cepheids of Lecture 07).

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📌 Key concepts

• Apparent vs. absolute magnitude: the latter removes the distance degeneracy — their difference is the distance modulus, $m-M=5{\log }_{10}d-5$.
• Color index: difference of magnitudes in two bands; a cheap, distance-independent proxy for effective temperature.
• OBAFGKM: decreasing temperature sequence; type A has the strongest H lines.

🔗 References and related

• Almeida-Fernandes et al. (2021) — photometric systems
• Perottoni et al. (2024) — photometric calibration (GaiaXPy)
• CursoON — overview
• Lecture 02 — HR Diagram & Star Clusters
• Lecture 04 — Spectroscopy & Metallicity
• Winter School — Galactic Archaeology, Lecture 01 — OBAFGKM classification revisited in a nucleosynthesis context (Portuguese only)