Questions: Photometric Magnitude Systems and Color Indices

Question 1

Star A has a B−V color index of −0.3. Star B has a B−V color index of +1.5. What does this tell you about their temperatures?

• A. Star A is cooler than Star B because its B−V index is smaller
• B. Star A is hotter than Star B because it is relatively brighter in blue than in visual wavelengths
• C. Star B is hotter because a larger positive color index indicates higher temperature
• D. The color index reveals nothing about temperature — you need a full spectrum for that

Answer: (B)

A negative B−V index means the star is brighter (smaller magnitude) in B (blue) than in V (green/visual), indicating it emits proportionally more blue light — the signature of a hot star. A large positive B−V (+1.5) means the star is much brighter in V than in B, indicating relatively more red/green light and a cool temperature. Hot O and B stars have B−V ≈ −0.3; cool M stars have B−V ≈ +1.5 or larger. The color index is essentially a measure of the slope of the spectral energy distribution — and temperature determines that slope via Wien's law.

Question 2

An astronomer reports that Star X has apparent magnitude +15 and Star Y has apparent magnitude +20. Which star is brighter and by approximately how much?

• A. Star Y is brighter because larger magnitude numbers indicate more light
• B. They are equally bright because the difference of 5 units is symmetric
• C. Star X is brighter by a factor of 5
• D. Star X is brighter by a factor of about 100

Answer: (D)

The magnitude scale runs backwards: smaller (or more negative) numbers mean brighter. A difference of 5 magnitudes corresponds to a flux ratio of exactly 100 (by the Pogson definition of the scale). Star X at magnitude +15 is 5 magnitudes brighter than Star Y at magnitude +20, so Star X is 100 times brighter in flux. This reversed, logarithmic scale is one of astronomy's most persistent sources of confusion — always remember: bright = small magnitude number.

Question 3

A star with apparent magnitude −1 is fainter than a star with apparent magnitude +6, because the negative magnitude indicates a smaller value.

• T. True
• F. False

Answer: False

The magnitude scale is inverted: smaller numbers (including negative numbers) mean brighter objects. Magnitude −1 is brighter than magnitude +6. Sirius at magnitude −1.46 is among the brightest stars in the sky, while magnitude +6 marks the faint limit of naked-eye vision. A difference of 7 magnitudes corresponds to a flux ratio of about 630. The historical origin is Hipparchus ranking stars 1 (bright) to 6 (faint), which the modern scale formalized into a logarithmic system — but preserved the counterintuitive direction.

Question 4

A color index is the difference between magnitudes measured in two different filters, and it carries information about the shape of a star's spectral energy distribution.

• T. True
• F. False

Answer: True

Because magnitudes are logarithmic, a magnitude difference corresponds to a flux ratio at two wavelengths. The ratio of fluxes at two wavelengths directly reflects the shape of the spectral energy distribution — which depends primarily on temperature. A B−V color index captures how the star's spectrum rises or falls between the B (445 nm) and V (551 nm) bands. Hot stars are brighter at shorter wavelengths (blue), giving negative B−V; cool stars are brighter at longer wavelengths (red), giving positive B−V. This is why color indices serve as temperature proxies without requiring spectroscopy.

Question 5

Explain why a V-band magnitude of 15.0 from the Johnson-Cousins system cannot be directly compared with a Sloan g′ magnitude of 15.0, and what information you would need to relate them.

Model answer: V and g′ are defined by different filter transmission curves that sample different wavelength ranges: the Johnson V filter is centered near 551 nm with a broad bandpass, while the Sloan g′ filter is centered near 469 nm with a different shape and reference standard. The same physical brightness (same number of photons per second per area) produces different magnitude values in each system because the filters weight different parts of the spectrum differently. A star's V = 15.0 and g′ = 15.0 would correspond to different flux levels. To convert between systems, you need transformation equations that account for the filter response curves — typically calibrated using stars measured in both systems. These transformations also depend on the star's color (spectral shape), since the offset between systems varies with temperature.

This is a practical issue that affects any large survey combining data from multiple instruments or epochs. Photometric calibration between systems is a non-trivial step in building sky catalogs. Without it, combining brightness measurements from different surveys introduces systematic errors proportional to the color difference of the stars being measured — redder stars are affected differently than bluer stars, meaning the error is not even a simple constant offset.