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A fundamental property of galaxy disks is their exponential or piece-wise exponential radial light profile (de Vaucouleurs 1959). Freeman (1970) noted that this profile gives a distribution of cumulative angular momentum versus radius that matches that of a flattened uniformly rotating sphere (Mestel 1963), but this match is only good for about four disk scale lengths. The problem is that an exponential disk has very little mass and a lot of angular momentum in the far-outer parts, unlike a power-law halo which has both mass and angular momentum increasing with radius in proportion (Efstathiou 2000). Nevertheless, observations show some disks with 8 to 10 scale lengths (Weiner et al 2001, Bland-Hawthorn et al 2005, Grossi et al 2011, Hunter et al 2011, Radburn-Smith et al 2012, Vlajić et al 2011, Barker et al 2012 , Hunter et al 2013, Mihos et al 2013, van Dokkum et al 2014). These large extents compared to the predicted four scale lengths from pure collapse models need to be explained (Ferguson and Clarke 2001).

Thus we have a problem: if the halo collapses to about four scale lengths in a disk, then how can we get the observed eight or more scale lengths in the stars that eventually form? The answer may lie with the conversion of incoming gas into stars. In a purely gaseous medium, interstellar collapse proceeds at a rate per unit area that is proportional to the square of the mass column density, Σgas (Elmegreen 2015). One factor of Σgas accounts for the amount of fuel available for star formation and the other factor accounts for the rate of conversion of this fuel into stars. This squared Kennicutt-Schmidt law converts four scale lengths of primordial gas into eight scale lengths of stars after they form (Sect. 7). Stellar scattering from clouds and other irregularities could extend or smooth out this exponential further (Elmegreen and Struck 2013, Elmegreen and Struck 2016).

There is an additional observation in Wang et al (2014) that in local gas-rich galaxies, the outer gas radial profiles are all about the same when scaled to the radius where ΣHI = 1 M pc−2. Bigiel and Blitz (2012) found a similar universality to the gas profile when normalized to R25, the radius at 25 magnitudes per square arcsec in the V band. Wang et al (2014) found that the ratio of the radius at 1 M pc−2 to the gaseous scale length in the outer disk is about four, the same as the maximum number of scale lengths in a pure halo collapse. This similarity may not be a coincidence (Sect. 7).

Cosmological simulations now have a high enough resolution to form individual galaxies with reasonable properties (Vogelsberger et al 2014, Schaye et al 2015). Zoom-in models in a cosmological environment show stellar exponential radial profiles in these galaxies (Robertson et al 2004) even though specific angular momentum is not preserved during the collapse and feedback moves substantial amounts of gas around, especially for low-mass galaxies (El-Badry et al 2016). For example, Aumer and White (2013) ran models with rotating halo gas aligned in various ways with respect to the dark matter symmetry axis. They found broken exponential disks with a break radius related to the maximum angular momentum of the gas in the halo, increasing with time as the outer disk cooled and formed stars. Star formation is from the inside-out. Angular momentum was redistributed through halo torques, but still the disks were approximately exponential. Aumer et al (2013) further studied 16 simulated galaxies with various masses. All of them produced near-exponential disks.

In a systematic study of angular momentum, Herpich et al (2015) found a transition from exponentials with up-bending outer profiles (Type III — Sect. 3) at low specific angular momentum (λ) to Type I (single exponential) and Type II (down-bending outer profiles outer parts) at higher λ. An intermediate value of λ = 0.035, similar to what has been expected theoretically (Mo et al 1998), corresponded to the pure exponential Type I. The reason for this change of structure with λ was that collapse at low spin parameter produces a high disk density in a small initial radius, and this leads to significant stellar scattering and a large redistribution of mass to the outer disk, making the up-bending Type III. Conversely, large λ produces a large and low-density initial disk, which does not scatter much and nearly preserves the initial down-bending profile of Type II.

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