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The luminous components in galaxies show a striking variety in morphology and in dimensions. Noticeably, the total luminosity and the radius R1/2 enclosing half of the latter are good tags of the objects.

3.1. Spirals, LSB and UDG

Caveat some occasional cases not relevant for the present topic, the stars are distributed in a thin disk with surface luminosity (Freeman 1970, for a study on 967 late type spirals, see Persic, Salucci and Stel 1996)

Equation 7


where RD = 1/1.67 R1/2 is the disk length scale, I0 is the central value of the surface luminosity and MD is the disk mass. The light profile of late spirals does not depend on galaxy luminosity and the length scale RD sets a consistent reference spatial scale. 2

The contribution to the circular velocity from this stellar component is:

Equation 8


where xR / RD and B = I0 K0I1 K1, a combination of known Bessel functions.

Classical LSB galaxies usually have central surface brightness down to µB(0) ∼ 22–23 mag arcsec−2 (Impey et al. 1988). Extremely low surface brightness (LSB) galaxies with unexpectedly large sizes, namely ultra-diffuse galaxies (UDGs), are found in nearby galaxy clusters (Bothun et al. 1991, Toloba et al. 2018). UDGs have much lower central surface brightness (µ(0) = 24–26 mag arcsec−2 in g band and half-light radii R1/2 > 1.5 kpc that, in spirals, are found in objects with stellar masses more than 10 times higher (van Dokkum et al. 2015, Shi et al. 2017). In LSBs/UDGs the stellar disks follow the Freeman exponential profile as in normal spirals, but their two structural parameters (I0 and RD) do not correlate as in the latter, where, approximately: LIRD2.

3.1.1. HI distribution in disk systems

Spirals have a gaseous HI disk which usually is important only as tracer of the galaxy gravitational field. Only at the outer radii (R > Ropt) of low luminosity objects, such disk becomes the major baryonic component of the circular velocity and must be included in the galaxy velocity model.

The HI disks show, very approximately, a Freeman distribution with a scale length about three times larger than that of the stellar disc (Evoli et al. 2011, Wang et al. 2014).

Equation 9


A rough estimate of the contribution of the gaseous disc to the circular velocity is

Equation 10


where the coefficient 1.3 is due to the He contribution. Of course when the resolved HI surface density is available, one derives VHI(R)2 directly from the latter. Inner H2 and CO disks are also present, but they are negligible with respect to the stellar and HI ones (Gratier et al. 2010, Corbelli and Salucci 2000).

3.2. Ellipticals

Ellipticals are more compact objects than spirals so that, in objects with same stellar mass M, they probe inner regions of the DM halo than spirals. Their profiles are well represented by the Sersic Law:

Equation 11


ΣS(0) = ΣRe eq, where R is the projected radial coordinate in the plane of the sky, ΣRe is the line of sight (l.o.s.) projected surface brightness at a projected scale radius ReR1/2 and q = 2m − 1/3 with m a free parameter. By deprojecting the surface density ΣS(R / Re, m), we obtain the luminosity density j(r) and by assuming a radially constant stellar mass-to-light ratio (M / L) we obtain the spheroid stellar density ρ(r).

3.3. Dwarf spheroids

The distribution of stars in dSph plays a major role in the analysis of their internal kinematics. The information we have comes from the bright stars detected by dedicated imaging or spectroscopy and, more recently, by surveys like the Sloan Sky Digital Survey and Gaia. The 3D stellar density is obtained from the deprojection of the 2D luminosity profile and an assumed mass-to-light ratio. The former is well reproduced by the Plummer density profile (Plummer 1915), characterized by a length scale Re and a central density ν0 = 3 Msph / (4π Re3) with Msph the total stellar mass. The projected mass (luminosity) distribution is given by: Σ(R) = [Msph / π Re2] (1 + x2)−2, x = R / Re. Then, the 3D stellar density is given by

Equation 12


2 We take Ropt ≡ 3.2 RD as the reference stellar disk edge. Back.

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