# What (actually) is the “ deprojected half-light radius” of this almost-all-dark-matter Galaxy?

The recent news of the Ultra Diffuse Galaxy (UDG) Dragonfly 44 is an excellent example of what could be termed 'observe different' thinking. The dragonfly telescope is noted not for the size of its collective aperture, but for the absence of the diffracting effects of secondary mirrors and surface roughness that limit the contrast of dim objects in conventional telescopes when brighter sources are nearby. See here and here and here.

above: image of a Dragonfly refractive array telescope from here. Image: P. Van Dokkum; R. Abraham; J. Brodie

above: The Dragonfly 44 ultradiffuse galaxy from here. "Dragonfly 44 is very faint for its mass and consists almost entirely of dark matter. (Pieter van Dokkum, Roberto Abraham, Gemini Observatory/AURA)"

Once identified, the radial velocities of the stars in Dragonfly 44 were measured using DEIMOS on the Keck II telescope, in order to determine a value for the mass for the dim, ultra-diffuse galaxy.

I started reading the ArXiv article but quicky got bogged down in the abstract. The very exciting result is that the luminosity and therefore total number of stars is much smaller than what one would expect from the mass obtained from the radial velocity measurements, suggesting that it is made almost entirely of dark matter. I wanted to see if I could understand how the mass was calculated, but I got stuck on the phrase deprojected half-light radius.

Could someone just outline how this calculation is done, and what that phrase actually means?

Recently a population of large, very low surface brightness, spheroidal galaxies was identified in the Coma cluster. The apparent survival of these Ultra Diffuse Galaxies (UDGs) in a rich cluster suggests that they have very high masses. Here we present the stellar kinematics of Dragonfly 44, one of the largest Coma UDGs, using a 33.5 hr integration with DEIMOS on the Keck II telescope. We find a velocity dispersion of 47 km/s, which implies a dynamical mass of M_dyn=0.7x10^10 M_sun within its deprojected half-light radius (my emphasis) of r_1/2=4.6 kpc. The mass-to-light ratio is M/L=48 M_sun/L_sun, and the dark matter fraction is 98 percent within the half-light radius. The high mass of Dragonfly 44 is accompanied by a large globular cluster population. From deep Gemini imaging taken in 0.4" seeing we infer that Dragonfly 44 has 94 globular clusters, similar to the counts for other galaxies in this mass range. Our results add to other recent evidence that many UDGs are "failed" galaxies, with the sizes, dark matter content, and globular cluster systems of much more luminous objects. We estimate the total dark halo mass of Dragonfly 44 by comparing the amount of dark matter within r=4.6 kpc to enclosed mass profiles of NFW halos. The enclosed mass suggests a total mass of ~10^12 M_sun, similar to the mass of the Milky Way. The existence of nearly-dark objects with this mass is unexpected, as galaxy formation is thought to be maximally-efficient in this regime.

## 2 Answers

The half light radius is the radius from within which half the luminosity emerges.

"Deprojected" means that the authors must have fitted some model to the 2D distribution of light, which can then be mathematically deprojected to give them a 3D model for luminosity as a function of radius, that they can then integrate to give a number for the half light radius.

In section 3, the authors explain that they have done this by fitting a "Sersic profile" to the surface brightness distribution https://en.m.wikipedia.org/wiki/Sersic_profile The Sersic profile actually has the 2D half light radius as one of its parameters. But if you imagine looking through a ball of stars, this 2D measurement of the half light radius is an underestimate of the true 3D half light radius, because the surface brightness profile is more sharply peaked than the 3D stellar density distribution that produces it.

The authors appear to approximately correct (deproject) this by multiplying the half light radius by 4/3. They also make a small correction for the non-sphericity of the galaxy.

The deprojection factor depends (slightly) on the $n$ index of the Sersic profile and needs to be found by doing a numerical integral. The details can be found in the appendices of Wolf et al. (2010. http://arxiv.org/abs/0908.2995 ), who also provide expressions to directly estimate mass from the projected half light radius and the line of sight velocity dispersion.

The half-light radius is the (spherical) radius from which half the electromagnetic power is radiated. Left unqualified it should mean the power in the entire electromagnetic spectrum, but it can also be constrained to cover a specific range of wavelengths. "Deprojected" has a straightforwrd meaning when considering a regular spiral galaxy. If you are viewing edge-on there will be more stars within a radius as viewed by you than if you were looking along the galaxies axis of rotation. Deprojected here means calculating what would be found if you were looking along the axis of the galaxy. For a spiral galaxy this radius is almost identical to the radius in the plane of the galaxy.
For other shapes, deprojection means working back to the 3D distribution of the stars based on a combination of measured data and a model of the galaxy that you are observing. (In my opinion the term deconvolving might be less confusing when dealing with bob-spherical 3D galaxies, but hisotry is what it is...)