Historical Perspectives on the Concept of Galaxy Size

Prior to the popularization of the effective radius (R_e is defined as the radius which encloses half the total light of a galaxy;³ de Vaucouleurs 1948) as a galaxy size measurement, it was common to refer to galaxy extensions as "diameter" or "boundary" in the literature. The apparent diameters of galaxies were initially discussed by Hubble (1926) and in later works "diameter" was defined using surface brightness isophotes. Redman (1936) was the first to suggest a "conventional" definition for the diameter of elliptical "nebulae." He proposed the use of the 25 Bmag/arcsec² isophote (D₂₅) for this purpose, using photographs of only five elliptical galaxies. The 25 Bmag/arcsec² "boundary" stemmed from two facts: (1) it was approximately the mean limiting depth (μ_lim) of the photographic plates and (2) it enclosed the bulk of a galaxy's light distribution.

A second measure for diameter emerged two decades later when Holmberg (1958) extracted microphotometer tracings of 300 "nebulae." He defined "diameter" as the radius in a galaxy where the photographic density with respect to the background in the plate is 0.5%. For operational purposes, however, Holmberg's radius (R_H) is measured at the location of the 26.5 Bmag/arcsec² isophote, i.e., close to μ_lim of Holmberg's plates.

Both D₂₅ and R_H operatively measured the maximum apparent boundaries of galaxies. But the use of "diameter," "dimension" or "boundary" interchangably in the literature to describe the concept of galaxy size was evident as early as the 1930s–40s (see e.g., Shapley 1942, and references therein). This was a period when long-exposure, homogeneous plates were used to compare the dimensions of spheroidal and spiral galaxies. The fact that several different terms referred to the same concept suggests that a clear framework to define exactly what "size" meant for a galaxy had not yet been realized. This is understandable: the outer regions of galaxies were poorly understood in that era and observations were highly biased toward the brightest regions of galaxies. Regardless of these issues in the 1930–40s, several astronomers were concerned about the right method of comparing the relative diameters or sizes of galaxies to study their growth and evolution. For instance, Shapley (1942) was well aware that only long-exposure, homogeneous data sets (i.e., deep imaging) can be used to compare the sizes of different galaxy populations. Furthermore, he recognized that if sizes of different populations are compared in the context of evolutionary scenarios, then the choice of the dimension parameter is extremely relevant.

Given the above circumstances, one may question when did astronomers start referring to parameters such as R_e, D₂₅ and R_H as size? At least in the case of R_e, this seems to have happened gradually, starting around the 1960s–70s, after the works by de Vaucouleurs (1959), Fish (1963) and Sérsic (1968). Fish (1963) was the first to demonstrate a correlation between R_e and luminosity for elliptical galaxies. He called this the "Luminosity Concentration Law": probably because R_e is a measurement of galaxy light concentration. However, even in Fish (1964) extensive study, R_e was never explicitly associated with a diameter or size definition for galaxies, but as a measurement of concentration and "radius" because it has units of length. Similar comments can be made about the first "mass–radius" relation using R_e by Sérsic (1968).

Interestingly, de Vaucouleurs (1959) can be considered the first attempt to compare the different measures we call size today, i.e., R_e and D₂₅. In his study, de Vaucouleurs (1959) generalized R_e to an "effective dimension," essentially placing it and the "brightness dimensions" on the same footing. This could be an early source for the confusion in the literature between what the original notion of dimension or diameter (as discussed in e.g., Hubble 1926; Redman 1936; Shapley 1942) meant and what R_e was tracing, i.e., light concentration (in e.g., Fish 1963). Subsequent comparisons were conducted by Vorontsov-Vel'Yaminov (1961) and Genkin & Genkina (1970) but have been ignored to date. In particular, Vorontsov-Vel'Yaminov (1961) is a critic against the use of R_e and isophotal definitions to compare galaxy sizes, pointing out that such a task "has no sense until the physical and practical determination of the boundaries of galaxies has been properly adjusted." And yet, although R_e and isophotal diameters were never introduced within a clear physical framework to define the luminous sizes of galaxies, they are used exactly for this purpose without any reserve.⁴

As current deep imaging has now allowed us to characterize the outskirts of galaxies with greater depth and accuracy than in the 1960s, it is time to regard critically our own conventions on galaxy size measures. In Trujillo et al. (2020), we proposed a new, physically motivated size definition based on the expected gas density threshold required for star formation in galaxies. In that study, we operatively selected a stellar mass density threshold as a proxy for such a definition. We chose the radius at the 1 M_⊙/pc² isomass contour (called R₁) because it is the density at the truncation in Milky Way-like galaxies. We measured R₁ for ∼1000 galaxies, from dwarfs to ellipticals, with stellar masses between 10⁷ M_⊙ < M_⋆ < 10¹² M_⊙. Compared to R_e, the new size measure not only captures the boundary of a galaxy, but also dramatically decreases the observed scatter in the size–stellar mass relation over the five orders of magnitude in M_⋆ by more than a factor of two. The need for a more representative measure of galaxy size is further supported in Chamba et al. (2020) where we demonstrated the misleading use of R_e to compare the sizes of galaxies. We studied a sample of ultra-diffuse galaxies (UDGs) and dwarfs and showed that the large R_e of UDGs does not imply that the galaxy is large in size, but only that their light distributions are less concentrated. Therefore, contrary to previous reports that UDGs are Milky Way-sized, our result shows that UDGs are 10 times smaller, similar to the sizes of dwarfs (Figure 1). We refer the reader to Trujillo et al. (2020) and Chamba et al. (2020) for more details.

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I thank Ignacio Trujillo and Johan H. Knapen for their support and comments. This project has received funding from European Union's Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement No. 721463 to SUNDIAL ITN.