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Sandbox Physics
Star cluster colours and populations

AC32 · L2

Two stars, one point

Sandbox Physics ·

A telescope receives light, not magnitudes. Add component light separately in each band, then use their shared position to calculate distance and selection. That chain produces one traceable point in a colour diagram.

Why averaging magnitudes fails

Replace one lamp with two identical lamps and the receiving surface gets twice the light. Flux adds directly. Magnitudes take a logarithm of flux so a large brightness range fits on a convenient numerical scale. Numbers that have already been logged cannot be added as though they were the original light.

In band bb, let fb,0f_{b,0} be the magnitude-system zero-point flux. Absolute magnitude MbM_b refers to placing the source at ten parsecs. Its flux ratio follows the definition below.

Mb=−2.5log⁡10(fb/fb,0),fb/fb,0=10−0.4MbM_b=-2.5\log_{10}(f_b/f_{b,0}),\qquad f_b/f_{b,0}=10^{-0.4M_b}
The logarithm is base ten. The zero point makes the flux ratio dimensionless; it does not make ratios in different bands interchangeable energies.
Mb,sys=−2.5log⁡10(10−0.4Mb,1+10−0.4Mb,2)M_{b,\rm sys}=-2.5\log_{10}\left(10^{-0.4M_{b,1}}+10^{-0.4M_{b,2}}\right)
For components at the same distance, add flux ratios under the same band zero point before returning to a magnitude. A single component contributes just one term.

Two identical components make a system approximately 0.752575 magnitudes brighter than either one. This number follows from the logarithmic definition, not a stellar-evolution parameter. Adding component birth masses does not justify looking up their summed mass as one single star. A single star with twice the mass has its own structure and luminosity.

Computed one-band magnitude shift as companion flux increases to equal primary flux
Own same-band flux arithmetic. Equal light gives a 0.752575 mag brightening. The horizontal coordinate is flux ratio, not stellar mass ratio or a native evolutionary model.

Reading: Kroupa (2001), On the variation of the initial mass function · Reyes et al. (2024), a precise M67 isochrone; native ISOA data

Finish each band before taking a colour

The Lab uses native G, BP and RP magnitudes. Look up each component’s three absolute magnitudes, combine the light within each band, and only then subtract RP magnitude from BP magnitude. Two BP flux ratios can be added, as can two RP ratios. BP and RP have different zero points, so adding their dimensionless ratios does not yield total physical energy.

For a checkable example, give the first component G, BP and RP absolute magnitudes of 4, 5 and 3, and the second 6, 7.5 and 5. The combined values are approximately 3.840270, 4.896518 and 2.840270. The resulting colour is approximately 2.056248, not the arithmetic mean of the component colours.

The public default takes magnitudes directly from 98 licensed native nodes. Local models interpolate mass only within a valid, equal-source-phase interval. Different phase labels refuse the bridge. If either component of a pair lacks support, the complete three-band system light is marked unsupported. That flag describes table support; the real component’s luminosity remains undetermined here.

Reading: Reyes et al. (2024), a precise M67 isochrone; native ISOA data · PARSEC/CMD3.8 official input help and phase labels · MIST official isochrone interface and user documentation

Distance belongs to the system’s position

A cluster has depth. Put an observer at a fixed direction behind its centre, and describe every system with three coordinates relative to that centre. All three enter its distance to the observer. Using the central distance for every component would discard the geometry we just assigned.

d=x2+y2+(D+z)2,mb=Mb,sys+5log⁡10(d/10 pc)d=\sqrt{x^2+y^2+(D+z)^2},\qquad m_b=M_{b,\rm sys}+5\log_{10}(d/10\,\mathrm{pc})
Coordinates and central distance use parsecs; the observer lies on the negative z axis behind the centre. There is no additional extinction term here: a local table may already include its selected extinction node, which must not be added twice.

This distance modulus follows from the inverse-square decline of received light. Moving the same system farther away adds the same magnitude shift in every band. Without extra extinction its colour stays fixed. Doubling distance adds approximately 1.50515 magnitudes. A brightness limit can then remove the system, changing the selected sample indirectly.

The record retains position, actual distance, component IDs and band light. The screen marker has compressed brightness and size so members remain visible. Distance always comes from the physical parsec coordinates.

Reading: Gaia DR3 documentation and archive

An aperture selects a cone

A telescope accepts an angular field. Different depths on the same line of sight therefore have slightly different allowed transverse radii. The green ring is the intersection with the cluster-centre plane. It does not turn the three-dimensional cone into a cylinder of fixed physical radius.

θ=atan2⁡ ⁣(x2+y2,D+z),θ<θap,G<18.5\theta=\operatorname{atan2}\!\left(\sqrt{x^2+y^2},D+z\right),\qquad \theta<\theta_{\rm ap},\quad G<18.5
A system must pass both angular and apparent-magnitude limits. A system must lie in front of the observer; a sample violating that geometric domain is refused.
Analytic continuum projected Plummer number CDFs for scales 1 and 2parsecs
Own deterministic continuum reference, separately normalized for each scale. It is not the finite seeded sample, its confidence band, a mass-weighted CDF or a dynamical age. Actual aperture selection uses each system depth.

A projected radius of 5 pc at a forward distance of 1000 pc subtends approximately 0.2865 degrees. An aperture of 0.2 degrees excludes it; 0.3 degrees includes it if the light also passes the magnitude limit. Neither operation moves the star or changes its age.

The radial chart uses projected physical radius from the cluster centre and separately accumulates supported system number and apparent G light. Its green line marks the aperture radius at the central plane. Actual selection still calculates every system angle. The line is not an exact physical-radius cut at every depth, and both cumulative curves refer only to systems with complete photometric support.

Reading: Polak et al., arXiv2408.14592v2 (16 September2025)

Keep one record all the way through

An observed synthetic point can now be traced back: which components it contains, which native mass or valid interpolation interval supplies their light, how G light adds, where the system lies, how far it is, and why it passes or fails selection. The scene, diagram and export carry the identities and calculated quantities of that same population.

These light predictions use several specific assumptions. Random pairing supplies no physical binary orbit or interaction. Concentration is imposed. A sharp magnitude limit is not a complete detection-efficiency model. Real data also need errors, field contamination and an extinction distribution. This model predicts what follows from stated assumptions; it does not invert the sky into parameter probabilities.

Reading: Reyes et al. (2024), a precise M67 isochrone; native ISOA data · Gaia DR3 documentation and archive

Exercises and explanations

  1. Two identical components each have absolute magnitude 5 in one band. What is the system magnitude?

    Read the explanation

    Approximately 4.247425 after adding flux. Adding the magnitudes would give 10, fainter than either component, contrary to the physical result of adding another lamp.

  2. Double the distance of one system without adding extinction. What happens to magnitude and colour?

    Read the explanation

    Every band becomes approximately 1.50515 magnitudes fainter, leaving their difference unchanged. A system near the brightness limit can disappear from the selected sample, so the sample colour distribution can change indirectly.

  3. Two systems have equal projected cluster radius but lie on the near and far sides. Must a fixed angular aperture select both together?

    Read the explanation

    No. The nearer system subtends a larger angle for the same projected radius and can lie outside while the farther system remains inside. Calculate each angle from the complete position.

Try the prediction

Trace one point to its components

  1. Export a population and find a detected system with two componentIds.
  2. Combine componentMagnitudes within each band, then use distancePc to predict apparentMagnitudes.

Change only the aperture

  1. Export a wide-field and narrow-field record.
  2. The same systems must retain identities, positions and band fluxes while inAperture and detected can change.

Sources and calculation scope

  1. Reyes et al. (2024), a precise M67 isochrone; native ISOA data

    The original ISOA file supplies one 3.95 Gyr age. Our CC BY 4.0slice uses98 monotonic rows with central hydrogen fraction above0.001; its native absolute magnitudes areGaia DR2, notEDR3.

  2. Kroupa (2001), On the variation of the initial mass function

    Read the distinction between a component mass distribution and unresolved systems. The finite0.1–8 solar-mass family in our local model is a stated teaching truncation, not a fittedM67IMF.

  3. PARSEC/CMD3.8 official input help and phase labels

    PinnedPARSEC 1.2S, solar-scaled composition, OBC andGaia EDR3Vega, native returned ages andAvnodes. Source labels are approximate; our equal-phase rule prevents bridging labelled boundaries but does not certify continuous evolutionary branches.

  4. MIST official isochrone interface and user documentation

    The local comparison usesMIST 1.2, MESA 7503, nonrotating, solar iron abundance, Av 0, and the originalGaia EDR3columns. Full original numerical hashes identify the twelve-node file; code licensing does not establish table redistribution permission.

  5. Gaia DR3 documentation and archive

    Astrometric reference epoch2016.0 is not a universal epoch for mean photometry. The tutorial uses a fixed cone and explicit astrometric cuts; source rows and real point plots are made only on the reader computer.

  6. Polak et al., arXiv2408.14592v2 (16 September2025)

    Sections2–4, Figures2–4 and AppendixA. Separate ongoing formation, dynamical rearrangement, ejections and wind-driven mass-rank changes; inspect which stars define each mass-segregation statistic.

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