Dwarf and giant galaxies

Ask which property is being compared

A galaxy can be described by its total luminosity, its physical extent, the distribution of its light and estimates of its stellar or total gravitating mass. Those are related measurements and inferences, but they are not interchangeable definitions of size. Calling an object faint does not specify whether it is compact or diffuse.

Dwarf and giant are useful comparative descriptors in galaxy research. A particular catalogue can adopt a boundary in a specified luminosity or mass measure, but there is no single numerical cutoff that can be moved unchanged among every band, population model and scientific question. State the convention when a boundary matters.

Surface brightness concerns light per apparent angular area on the sky. Two objects with equal integrated received flux can distribute it across different angular areas. Their observed profiles and the observing system’s response then affect how clearly that light stands above a background.

A stellar-mass estimate additionally requires population assumptions such as those in the previous lesson. A dynamical inference concerns gravitating matter under an adopted model. Low luminosity alone establishes neither a unique stellar mass nor the amount or identity of an unseen component.

Spread a fixed source across two apertures

Consider two original, hypothetical detected images. Each has expected total source signal S = 120 electron counts in the chosen exposure. In one image that signal is uniformly spread over N = 4 equal angular-area pixels; in the other it is uniformly spread over N = 16 such pixels. Both apertures enclose all of the supplied source signal.

The expected background is b = 30 electron counts per pixel in both images. Suppose source and background counts follow independent Poisson statistics: each contribution has count variance equal to its expected count. Treat the mean background as perfectly known and subtracted. Subtracting that mean does not remove the random fluctuations of the background photons already detected.

This deliberately optimistic model omits read noise, dark-current uncertainty, correlations, calibration errors and uncertainty in estimating the background. It also supplies uniform, resolved images rather than real galaxy profiles convolved with an instrumental response. The calculation isolates the angular-footprint penalty under these assumptions.

The mean source per pixel is S/N, giving 30 counts for the four-pixel image and 7.5 for the sixteen-pixel image. The total expected source is unchanged. More aperture pixels, however, collect more of the stipulated background while summing that same source signal.

Add variances before taking the square root

The summed background expectation is Nb. Independent Poisson contributions give summed count variance S + Nb. The standard deviation is therefore σ = √(S + Nb), in the corresponding count units, and the conditional integrated signal-to-noise ratio is S/√(S + Nb). The variance adds before the square root is taken.

For four pixels, Nb = 120, the variance is 240 and S/σ = 120/√240 = √60 ≈ 7.745966692. For sixteen pixels, Nb = 480, the variance is 600 and S/σ = 120/√600 = √24 ≈ 4.898979486. The supplied total source did not fall; the background variance increased.

These are ratios computed from expected counts, not particular measured images or probabilities of detection. Whether a survey admits an object depends on its detection procedure and other uncertainties as well as any adopted criterion. The table supplies neither a universal significance threshold nor the probability that either object enters a real catalogue.

A point-spread function, overlapping sources, sky structure and imperfect background subtraction can complicate the comparison. The STScI count-noise discussion supports the distinction between a subtracted mean background and remaining background noise; this exercise retains the explicit known-background assumption.

A label does not replace a physical investigation

Some low-luminosity systems raise questions about the boundary between galaxies and star clusters. A cluster of stars and a galaxy can overlap in a selected size or luminosity measure, so one image or one catalogue descriptor need not settle the physical interpretation.

Willman and Strader discuss proposed evidence-based criteria for what should count as a galaxy, including dynamical and chemical evidence. Their proposal should be presented as a proposed operational definition, rather than a universally adopted rule or a conclusion that every object with a particular radius must satisfy.

Interpreting a small system can require member identification, motions, chemical abundances and attention to contamination, binary stars and departures from the adopted dynamical assumptions. A measured spread or unusually large inferred mass-to-light ratio must be connected to the actual data and model before it supports a physical claim.

Galaxy populations also contain diversity in structure, gas content and stellar history. Dwarf is not a synonym for young, irregular, gas-rich or dark-matter dominated under every measurement convention. An empirical tendency in a selected sample is different from a definition that applies to every member.

Which faint systems could the survey have found?

A catalogue can miss an extended low-surface-brightness source even when its integrated luminosity would seem adequate. Detection, separation of overlapping images, background modelling, target choice and successful follow-up can each change which objects appear in the final sample.

Tests that inject artificial objects into data can measure recovery for the supplied profiles, locations, angular sizes and observing conditions. Such tests are valuable conditional evidence. They do not prove completeness for every unknown kind of source outside the tested templates or for a different follow-up selection.

Blanton and collaborators discuss how low-surface-brightness selection affects the local galaxy population inferred from survey data. A stated luminosity function therefore needs an observing band, distance treatment, footprint, selection description and completeness assumptions. Missing diffuse systems cannot be repaired simply by renaming the detected ones.

The next lesson turns from individual galaxy properties to groups, clusters and environment. The same habit applies: define the measured quantity and the selection rule before treating a catalogue category as a complete physical population.

Worked examples

Try the reasoning

1. Does low total luminosity alone imply a small physical radius or low dynamical mass?

   Show the answer and reasoning

   No.

   Extent, stellar population and gravitating mass require additional measurements and assumptions.

2. Does perfect knowledge of the mean background eliminate the fluctuations of detected background counts?

   Show the answer and reasoning

   No.

   It eliminates uncertainty in that mean in this model; background Poisson variance remains.

3. Is the supplied expected-count signal-to-noise ratio itself a catalogue admission probability?

   Show the answer and reasoning

   No.

   A detection probability requires the actual selection procedure and a statistical model for its outcomes.

4. Does recovery of one injected source family establish completeness for every unknown galaxy profile?

   Show the answer and reasoning

   No.

   Recovery tests are conditional on their templates, conditions and subsequent selection steps.

What to carry forward

• Luminosity, surface brightness, extent and inferred mass describe different properties.
• In the supplied Poisson model, spreading fixed source counts over more background pixels reduces integrated signal-to-noise.
• Classification and completeness statements need explicit evidence, definitions and selection assumptions.