Astronomy · Comparing galaxies

Light, colour and mixed stellar populations

Add two declared population contributions before calculating colour, and compare current stellar-mass fractions with band-dependent light fractions.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Distinguish mass weighting from luminosity weighting in a composite galaxy.
Calculate total band luminosities and a calibrated colour from a supplied mixture.
Identify model assumptions needed before interpreting an integrated colour as population history.

One spectrum can contain many stellar populations

The unresolved light of a galaxy combines contributions from many stars with different physical properties and histories. In a specified band, the luminosities of those components add. An integrated colour is formed from calibrated band totals after that addition; it is not generally the arithmetic average of the components’ magnitudes or colours.

Stellar population synthesis connects a proposed mixture of stellar ages and compositions to a predicted spectrum or band luminosities. It requires evolutionary models, stellar spectra and a population accounting convention. A galaxy need not be one single-age, single-composition population.

The current stellar mass is also a specified quantity. It can include living stars and remnants under a declared convention, while excluding returned gas. A model normalized per unit mass originally formed cannot be substituted without conversion for one normalized per unit current retained stellar mass.

Dust attenuation and emission, gas emission and an active nucleus can add further complications to an observed spectrum. The following original ledger isolates stellar light addition with two deliberately invented components; it does not fit a real galaxy or identify either component with a particular age.

Fix the normalization before changing the mixture

At one chosen epoch, stipulate total current stellar mass M_A + M_B = 100M0, where M0 is an arbitrary positive mass unit. Here current stellar mass includes living stars plus remnants and excludes gas. Let f_B = M_B/(100M0), so M_A/M0 = 100(1 − f_B) and M_B/M0 = 100f_B.

Use fictional fixed bands X and Y and an arbitrary common power unit L0 for their integrated band luminosities. Per M0 of current mass, component A supplies (L_X, L_Y) = (1,2)L0 and B supplies (20,10)L0. These are mathematical templates, not actual stellar spectra, filter curves or population-model outputs.

Assume transparent propagation, isotropic emission and a common observer distance for this comparison. Define calibrated toy colour C = −2.5 log₁₀[(L_X/L0)/(L_Y/L0)]. The equal numerical reference in the two bands is stipulated, rather than borrowed from a real magnitude system. Neither band luminosity is a bolometric luminosity.

All five mixtures below share the same total current mass and the same component templates. They are separate hypothetical mixtures, not five times along an evolution track. No changes in mass return, formation rate, dust or distance are included.

Original abstract luminosity templates per M0 of current stellar mass. L0 is a common arbitrary power unit for the two fictional band integrals. No ages, real filters or stellar spectra are supplied.
Component	L_X per M0, in L0	L_Y per M0, in L0
A	1	2
B	20	10

A small mass fraction can dominate a band

Multiplying each template by its component’s current mass gives L_X,A/L0 = 100(1 − f_B) and L_X,B/L0 = 2000f_B. The total is L_X/L0 = 100(1 + 19f_B). Similarly, L_Y,A/L0 = 200(1 − f_B), L_Y,B/L0 = 1000f_B and L_Y/L0 = 100(2 + 8f_B).

At f_B = 0.10, A contains 90M0 and B contains 10M0. Their X luminosities are 90L0 and 200L0, giving total 290L0. Their Y luminosities are 180L0 and 100L0, giving total 280L0. B has only one tenth of the current mass but contributes 20/29 of the X light and 5/14 of the Y light.

The resulting colour is C = −2.5 log₁₀(290/280) ≈ −0.038099916 magnitudes. The light fraction that dominates depends on the band: B dominates X in this mixture but not Y. A statement about which component is brightest therefore needs its observing response.

The supplied formulas permit every row to be reproduced from component contributions. A changing colour in this ledger tracks the stipulated mixture, not an independently measured temperature, age, metallicity or formation rate.

Five original current-mass mixtures at one epoch. Band totals follow the explicit component-sum formulas. Light fractions and calibrated toy colour are rounded; these are not observed galaxies or a time sequence.
f_B	M_A/M0	M_B/M0	L_X/L0	L_Y/L0	B fraction of X light	B fraction of Y light	C (mag)
0	100	0	100	200	0	0	0.752574989
0.01	99	1	119	208	0.168067227	0.048076923	0.606290934
0.05	95	5	195	240	0.512820513	0.208333333	0.225441576
0.10	90	10	290	280	0.689655172	0.357142857	−0.038099916
0.20	80	20	480	360	0.833333333	0.555555556	−0.312346842

Light weighting answers a different question

A luminosity-weighted summary gives greater influence to components supplying more of the measured light. A mass-weighted summary uses their mass contributions instead. They can differ substantially without either calculation being arithmetically wrong; they summarize different aspects of the population.

For the f_B = 0.10 mixture, the current stellar mass-to-X-light ratio is (100/290)M0/L0 = (10/29)M0/L0. That number uses the declared current stellar mass and band X. It is not a ratio for all gravitating mass and cannot be moved unchanged to band Y or to a birth-mass convention.

Real population inference also depends on the initial mass distribution, stellar lifetimes, remnant accounting, composition and formation history. The part of a population that dominates a selected band need not dominate mass, and the numerical relation between those contributions is model-dependent.

Conroy’s review and Bovy’s population-synthesis treatment discuss these connections among component spectra, population histories and retained mass. The original ledger exposes the weighting issue without importing a universal colour-to-mass calibration or assigning a fixed age to either abstract component.

A colour is a constraint, not a unique history

A redder integrated colour can arise from a changed stellar mixture, attenuation or a combination of effects. Emission from gas or a compact nucleus can alter selected bands as well. Treating every change as stellar ageing would discard those alternatives.

More bands and spectra can constrain different parts of a model, while spatially resolved measurements can help separate regions or a bright centre. Their usefulness depends on calibration, uncertainty and whether the chosen features distinguish the competing explanations. More measured numbers do not automatically remove every degeneracy.

Comparisons should therefore state the bands, mass convention, attenuation treatment and population assumptions. A fit that reproduces total colours can still leave uncertain formation histories or contributions from faint stars and remnants. Integrated light alone supplies no unique dark-matter estimate.

The next lesson compares dwarf and giant descriptors and the detectability of diffuse light. Keeping total luminosity, luminosity per angular area and inferred mass distinct will be essential there.

Worked examples

Calculate the luminosity shares separately

At f_B = 0.10, what fraction of each band’s light comes from B?

In X, A contributes 90L0 and B contributes 200L0, so B’s fraction is 200/290 = 20/29.
In Y, A contributes 180L0 and B contributes 100L0, so B’s fraction is 100/280 = 5/14.
Compare those light fractions with the supplied current-mass fraction 0.10; neither band is weighted by mass alone.

B supplies about 0.689655172 of X light and 0.357142857 of Y light while holding 0.10 of current stellar mass.

Apply the calibration to the composite totals

What toy colour does the same mixture have, and what physical property has that calculation established?

Sum the two components to obtain L_X = 290L0 and L_Y = 280L0.
Use the declared equal-reference calibration: C = −2.5 log₁₀(290/280).
The result is approximately −0.038099916 magnitudes; the supplied templates have no assigned ages or stellar temperatures.

The colour is approximately −0.038099916 mag. It describes this stipulated composite spectrum through two fictional bands and does not uniquely establish an age or temperature.

Try the reasoning

Should component colours generally be averaged directly to obtain the unresolved system’s colour?
Show the answer and reasoning
No.
Add calibrated component band fluxes or luminosities first, then form the colour from the totals.
Does B’s dominance in X at f_B = 0.10 imply dominance in Y or in current stellar mass?
Show the answer and reasoning
No.
The supplied Y light fraction is 5/14 and the current-mass fraction is 0.10.
Is the supplied current stellar mass-to-light ratio automatically a total dynamical mass-to-light ratio?
Show the answer and reasoning
No.
Its numerator excludes gas and any other gravitating component outside the stated stellar-mass convention.
Does one integrated colour uniquely determine a galaxy’s formation history?
Show the answer and reasoning
No.
Population mixture, attenuation, calibration and possible nonstellar emission require additional constraints and modelling.

What to carry forward

Band luminosities add before a composite colour is calculated.
Current stellar-mass fractions and band luminosity fractions answer different questions.
Population interpretation needs declared mass, spectral and attenuation assumptions alongside additional evidence.

References

Teaching resources and research supporting this lesson’s explanations and examples.