# What are isophotal diameters? And how do they relate to galaxies' size?

I heard that isophotal diameters are a common way to measure a galaxy diameter. How do they work, and what relationship is there between the two values?

• What do you mean by a galaxy's "real" diameter? That seems as undefinable as the "real" height of the Earth's atmosphere or the "real" radius of the solar system. Some things are nebulous and simply don't have a "true", objective, absolute, well-defined size. Commented Jul 27 at 15:48
• Yeah I mean, that is what I also think, but I surely have said that badly. With "real" I wanted to mean the part of a galaxy that we usually associate with the term, e.g. the disk for a spiral. Again, my fault, thanks for noticing Commented Jul 27 at 16:33
• "I heard that..." if there's any way to add more context about where or how you herd it - especially if you can link to it, it will be helpful to potential answer authors.
– uhoh
Commented Jul 27 at 21:41
• @Miss_Understands "The outermost closed isophote loop is the diameter" -- no, the diameter is defined by the isophote corresponding to a particular value of the surface brightness. The most common definition is 25 magnitudes per square arc second in the B band, but there are other definitions using other values in other filters/wavelengths. Commented Jul 27 at 23:08
• @Miss_Understands Perhaps you could convert your comment (minus the aggressive tone) and add something along the lines that Peter Erwin mentioned. Commented Jul 28 at 17:40

Galaxies are intrinsically fuzzy objects that usually fade away fairly smoothly in density as you go out in radius. So there isn't really such a thing as the diameter or size.

Isophotal diameters are an attempt to come up with a standard (and simple) way of measuring galaxy sizes in ways that can be compared between different galaxies. It basically means the diameter or radius (along the galaxy's major axis if it's not circular) at which the integrated stellar brightness has declined down to some standard value. The most popular such value is 25 magnitudes per square arc second in the B band. Other definitions exist: for example, there's the once-popular "Holmberg radius" (26.5 mag arcsec$$^{-2}$$, also in the B band), and the Spitzer Survey of Stellar Structure in Galaxies (Sheth et al. 2010) uses both 25.5 and 26.5 mag arcsec$$^{-2}$$ in the Spitzer 3.6 micron band.

Here's an attempt to illustrate this for the S0 galaxy NGC 3412. These are two $$r$$-band images I obtained back in 2004 with the Isaac Newton Telescope's Wide Field Camera (Erwin et al. 2008). I've plotted their isophotes (after smoothing the images with a median filter) using steps of 0.5 mag arcsec$$^{-2}$$. I've converted both to B-band assuming a constant color of $$B - R = 1.46$$; since this is an S0 galaxy with no recent star formation, assuming one color for the entire galaxy is not too terrible an approximation.

The first image is a 30-second exposure; the signal-to-noise is so low that I stop plotting isophotes at the level of 26.5 mag arcsec$$^{-2}$$.

The second image is a combination of several exposures totalling 2400 seconds (you can see that it's actually saturated in the center); the faintest plotted isophote is 28.5 mag arcsec$$^{-2}$$.

In both plots, I've outlined the 25.0 mag arcsec$$^{-2}$$ isophote in blue. This is pretty elliptical (a bit ragged in the "30s" plot since the S/N is so low) and has a semi-major-axis of $$\sim 98$$ arc seconds, for a "diameter" of $$\sim 200$$ arc seconds. At the 11 Mpc distance of NGC 3412, this translates to a diameter of $$\sim 10.7$$ kpc, or about 35,000 light years.

(For comparison, the RC3 25 mag arcsec$$^{-2}$$ diameter for this galaxy is 218 arc seconds, measured on photographic plates.)

• Perfect, thanks for clearing things up. An answer from an expert in the field is always the best one can look for. Commented Jul 29 at 13:38