Accretion onto disk galaxies via hot and rotating CGM inflows

Observed accretion rates onto the Milky-Way and other local spirals fall short of that required to sustain star formation for cosmological timescales. A potential avenue for this unseen accretion is an inflow in the volume-filling hot phase ($\sim10^6$ K) of the circumgalactic medium (CGM), as suggested by some cosmological simulations. Using hydrodynamic simulations and a new analytic solution valid in the slow-rotation limit, we show that a hot inflow spins up as it approaches the galaxy, while remaining hot, subsonic and quasi-spherical. At the radius of angular momentum support ($\approx15$ kpc for the Milky-Way) the hot flow flattens into a disk geometry and then cools from $\sim10^6$ K to $\sim10^4$ K at the disk-halo interface. Cooling affects all hot gas, rather than just a subset of individual gas clouds, implying that accretion via hot inflows does not rely on local thermal instability in contrast with 'precipitation' models for galaxy accretion. Prior to cooling and accretion the inflow completes $\sim t_{\rm cool}/t_{\rm ff}$ radians of rotation, where $t_{\rm cool}/t_{\rm ff}$ is the cooling time to free-fall time ratio in hot gas immediately outside the galaxy. The ratio $t_{\rm cool}/t_{\rm ff}$ may thus govern the development of turbulence and enhancement of magnetic fields in gas accreting onto low-redshift spirals. We argue that accretion via hot inflows can explain the observed truncation of nearby thin stellar disks at $\approx4$ disk radii. We also show that if rotating hot inflows are common in Milky-Way size disk galaxies, as predicted, then signatures should be observable with X-ray telescopes and FRB surveys.


INTRODUCTION
Observation of neutral gas surrounding the Milky Way and nearby spirals suggest accretion rates of 0.05 − 0.2 M ⊙ yr −1 , falling short of the 1 − 2 M ⊙ yr −1 required to sustain observed star formation rates (SFRs) for cosmological timescales (Sancisi et al. 2008;Putman et al. 2012;Kamphuis et al. 2022).This 'missing accretion' is often attributed to predominantly ionized gas clumps with temperature ∼ 10 4 K (e.g., Voit et al. 2017), observable mainly in UV absorption.It is however unclear if this phase can provide the necessary fuel for star formation, due to both uncertainties in converting UV absorption features to net accretion rates (e.g.Fox et al. 2019), and since hydrodynamic instabilities may disrupt and evaporate cool gas clumps before they reach the galaxy (Heitsch & Putman 2009;Armillotta et al. 2017;Tan et al. 2023;Afruni et al. 2023).An alternative, less explored possibility is that accretion proceeds via a subsonic inflow in the volume-filling hot phase (∼ 10 6 K) of the circumgalactic medium (CGM), similar to classic 'cooling flow' solutions discussed in the context of the intracluster medium (ICM, Mathews & Bregman 1978;Fabian et al. 1984).Such hot CGM inflows are evident in modern cosmological simulations such as FIRE (Stern et al. 2021a;Hafen et al. 2022) and TNG (ZuHone et al. 2023; see also figure 9 in Nelson et al. 2019).
Since the hot CGM is expected to have a net rotation (e.g., Roškar et al. 2010;Stevens et al. 2017;Oppenheimer 2018;DeFelippis et al. 2020;Truong et al. 2021;Huscher et al. 2021;Nica et al. 2021), an inflow will cause it to spin up.Stern et al. ( 2020) used an idealized one-dimensional model to show that in Milky-Way mass halos, such a rotating hot inflow will remain hot down to the radius where the rotation velocity approaches the circular velocity  c = √︁   (< )/, at which point the gas cools to ∼ 10 4 K and joins the ISM disk.Hafen et al. (2022) demonstrated that this picture applies, and is the dominant accretion mode onto  ∼ 0 Milky-Way mass galaxies in the FIRE-2 cosmological zoom simulations (Hopkins et al. 2018).They further showed that the flow forms a coherently spinning disk prior to accretion onto the galaxy, and that this coherence may be a necessary condition for the formation of thin disk galaxies, consistent with conclusions from related FIRE-2 analyses (Stern et al. 2021a,b;Yu et al. 2021Yu et al. , 2022;;Gurvich et al. 2023).It thus appears that a deep understanding of the physics of hot and rotating inflows could be crucial for understanding the evolution of local star forming disks.
In the present paper, we complement the cosmological simulationbased analysis of hot and rotating CGM inflows in Hafen et al. (2022), by deriving an idealized, two-dimensional axisymmetric solution for inflowing and rotating hot CGM.Deriving an idealized solution allows identifying its dependence on system parameters and boundary conditions, and provides a basis for assessing the effects of additional physics.Our derivation is built on previous 1D hot inflow solutions 1 which accounted for rotation in a highly approximate manner (Cowie et al. 1980;Birnboim & Dekel 2003;Narayan & Fabian 2011; Stern 1 Also called 'cooling flows', though note this term has been used also for flows in which the gas does not remain hot (e.g., McQuinn & Werk 2018).et al. 2020).These 1D studies assumed the centrifugal force is directed outward in the spherical radius direction, so the solution remained spherically symmetric.Here we assume the centrifugal force is directed outward in the cylindrical radius direction, and derive a 2D axisymmetric solution which captures the transition from a quasi-spherical flow at large scales where rotational support is weak to a disk geometry at small scales where rotational support dominates.The idealized nature of our approach implies that insights may be applicable also to other astrophysical disks fed by spherical inflows, such as AGN disks in galaxy centers (e.g., Quataert & Narayan 2000) or protoplanetary disks in the center of star-forming clouds (e.g., Fielding et al. 2015).
The inflowing hot CGM solution derived herein focuses on the limit where feedback heating is subdominant to radiatively cooling, thus differing qualitatively from radially-static hot CGM models (also known as 'thermal balance' models, e.g., McCourt et al. 2012;Sharma et al. 2012b;Voit et al. 2017;Faerman et al. 2017Faerman et al. , 2020;;Pezzulli et al. 2017;Sormani et al. 2018), and from hot outflow models (Thompson et al. 2016;Schneider et al. 2020).Thermal balance models assume that radiative cooling is equal to feedback heating, thus inhibiting the hot inflow, while outflow models require that feedback heating dominates.Observational evidence for thermal balance is strong in the ICM, since the star formation rate (SFR) at the cluster center is small relative to the inflow rate  implied by the X-ray emission   .This is the well-known 'cooling flow problem', where  ≈   / 2 c ≫ SFR (see McDonald et al. 2018 for a recent revisit).There is, however, no similar cooling flow problem in disc galaxies.Upper limits on   from the hot CGM of Milky-Way mass galaxies are a few ×10 40 erg s −1 (Li & Wang 2013;Li et al. 2014;Anderson et al. 2015;Comparat et al. 2022), and recent results based on eROSITA data indicate the actual emission may be comparable to this value (Chadayammuri et al. 2022).For  c ≈ 200 km s −1 this   implies  ≈ 1 M ⊙ yr −1 ∼ SFR, in contrast with  ≫ SFR deduced for the ICM.Similarly, estimates of  in the Milky-Way halo suggest  ≈ 0.1−1 M ⊙ yr −1 (Miller & Bregman 2015;Li & Bregman 2017;Stern et al. 2019), again inconsistent with the  ≫ SFR derived for the ICM.More massive spirals in which the hot CGM is detected in individual objects with   ≳ 10 41 erg s −1 have SFR ≈ 10 M ⊙ yr −1 and hence also satisfy  ∼ SFR (Anderson et al. 2016;Bogdán et al. 2017;Das et al. 2019).A cooling flow problem akin to that in the ICM does not exist in the CGM of disk galaxies, allowing for the possibility that the hot CGM is inflowing.
The paper is organized as follows.In section 2 we discuss the structure and properties of hot and rotating CGM using analytic considerations, while in section 3 we derive a numerical solution.In section 4 we consider the effect of additional physical mechanisms which were not included in the basic analysis, and in section 5 we derive several observables of hot rotating CGM.Implications of our results are discussed in section 6 and section 7 provides a summary.

THE STRUCTURE OF HOT AND ROTATING CGM -ANALYTIC CONSIDERATIONS
The flow equations for radiating, ideal gas with adiabatic index  = 5/3 subject to an external gravitational potential Φ are where ,  and ì  are respectively the gas density, pressure, and velocity.We use  ≡  −5/3 for the 'entropy' (up to an exponent and a constant) and  cool for the cooling time, defined as where  H is the hydrogen density, (3/2) is the energy per unit volume, and Λ is the cooling function defined such that  2 H Λ is the energy lost to radiation per unit volume per unit time.Equations ( 1)-(3) neglect conduction, viscosity and magnetic fields, the potential effect of which will be assessed below.We also do not include a heating term in equation (3) since we search for a solution in the limit that heating is subdominant to cooling (see introduction).

Hot CGM without angular momentum
We start with a brief review of steady-state (/ = 0) hot inflow solutions without angular momentum, which were studied extensively mainly in the context of the inner ICM (classic 'cooling flows', e.g., Mathews & Bregman 1978) and adapted to galaxy-scale halos by Stern et al. (2019).When angular momentum is neglected spherical symmetry can be assumed, and hence eqns.(1)-(3) reduce to where  is the spherical radius,  is the mass flow rate (constant with radius in steady-state, down to the radius where stars form),  turb =  2 turb is the turbulent pressure, and  turb is the turbulent velocity.Multiplying both sides of eqn.( 7) by the free-fall time This equation indicates that if  cool ≫  ff , then either the flow is isentropic with  ln / ≈ 0 as in the Bondi (1952) solution, or that the inflow velocity is small, i.e., The solutions discussed here correspond to the latter type of solutions where   ≪  c , known as cooling flow solutions in the ICM literature.Hydrodynamic simulations show that initially static gas with  cool ≫  ff converges onto a cooling flow solution within a timescale  cool , rather than onto an isentropic flow (e.g., Stern et al. 2019).
To derive an analytic approximation, one can neglect in eqn.( 6) the small inertial term  2  and the turbulent term which is also expected to be small  turb ∼ (  / c ) 2  th (see section 4.2 below).Further approximating the gravitational potential as isothermal with some constant  c then gives (see section 2 in Stern et al. 2019): where  s is the sound speed,  is the temperature, M  ≡   / s is the radial Mach number of the flow, and we normalized by the following numerical values: The numerical values used in eqn.( 10) are appropriate for the Milky Way CGM:  is taken to be roughly half the star formation rate (SFR) of ≈ 1.5 − 2 M ⊙ yr −1 (Bland-Hawthorn & Gerhard 2016), as expected in steady-state where the ISM mass is constant with time, and ≈ 40% of the stellar mass formed is ejected back into the ISM via winds and supernovae (e.g.Lilly et al. 2013).This  is also consistent with X-ray absorption and emission constraints on the hot CGM of the Milky-Way (see introduction).The value of Λ is appropriate for  = 2 • 10 6 K gas with metallicity  ⊙ /3, as measured for the Milky-Way CGM (Miller & Bregman 2015).
Eqn. (10) reveals several properties of the non-rotating solution.The inflow velocity   is equal to / cool , so the accretion time equals the cooling time, as expected in a cooling flow.This also implies that the entropy drops linearly with decreasing radius (see eqn. 7).Additionally, the inflow has a temperature which is independent of radius and roughly equal to the halo virial temperature, despite radiative losses.This is a result of compressive heating during the inflow balancing radiative cooling.
The solution in eqn.(10) also highlights that the parameter  cool / ff sets the Mach number of the flow, and thus also the sonic radius of the flow where |M  | = 1: where we approximated √︁ 9/20 ≈ 0.7.Note that near and within the sonic radius the assumption of a quasi-hydrostatic flow is invalid, so the estimate of  sonic is approximate.Equation (11) indicates that  sonic is well within the galaxy for Milky-Way parameters, though it can be on CGM scales in lower mass galaxies where  c is lower, or at higher redshift where  is higher.In this paper we focus on systems with  sonic smaller than the galaxy scale, so the quasi-hydrostatic approximation is valid throughout the CGM.
Another important scale of cooling flows is the cooling radius  cool at which  cool equals the system age or the time since the last heating event.This scale is not part of the steady-state solution, since a cooling flow develops on a timescale  cool and thus steadystate is achieved only at  <  cool .A time-dependent solution for how the outer boundary of the inflow at  ∼  cool expands as  cool grows was derived by Bertschinger (1989).For the above parameters  cool = 10 Gyr occurs at  = 110 kpc, so when necessary we assume  cool = 100 kpc.The present study however focuses on smaller radii of  ≲ 40 kpc where the dynamical effects of angular momentum are most pronounced, so  cool is short relative to cosmological timescales and thus steady-state is more likely to be achieved.This inner CGM region is also less susceptible to cosmological effects not included in our analysis, such as non-spherical accretion and satellite galaxies (Fielding et al. 2020).

Rotating hot CGM inflows -the circularization radius
Given some net angular momentum in the hot gas, for example due to torques applied by neighboring halos, the rotation velocity will increase as the gas inflows.We can hence define a circularization radius  c as the radius where the rotational velocity equals the circular velocity and the flow becomes rotationally supported: where here and henceforth we use  for the cylindrical radius (and  for the spherical radius).One can also express  c using the specific angular momentum of the hot gas : Cosmological considerations can be used to estimate a typical  c .In a ΛCDM universe, a given dark matter halo is expected to have a spin parameter  which on average equals: (e.g., Bullock et al. 2001;Rodríguez-Puebla et al. 2016), where  vir ,  vir ,  vir , and  correspond to the halo mass, virial velocity, virial radius, and angular momentum, correspondingly.Assuming the hot CGM has roughly the same spin as the dark matter halo as suggested by cosmological simulations (e.g., Stewart et al. 2013Stewart et al. , 2017;;DeFelippis et al. 2020;Hafen et al. 2022), and assuming that near the disk  c =   c  vir with   c ≳ 1 we get Comparison of eqn.(15) with eqn.(11) implies that  c ≫  sonic in Milky-Way halos.Thus, a hot CGM inflow in a Milky-Way halo is expected to become rotation-supported well before it transitions into a supersonic flow.This conclusion is also apparent if we estimate the radial Mach number near  c .Using eqn.(10) we have and hence The difference between CGM with  sonic <  c and CGM with  sonic >  c was discussed by Stern et al. (2020), and is related to the classic distinction between 'hot mode' and 'cold mode' accretion (White & Rees 1978;Birnboim & Dekel 2003;Fielding et al. 2017).
In this paper we focus on systems with  sonic <  c so the hot accretion mode dominates.

Rotating hot CGM inflows -fluid equations and outer boundary condition
Accounting for angular momentum, and assuming steady-state and axisymmetry, the  and  components of the momentum equation ( 2) reduce to where  is the angle relative to the rotation axis and Ω =   /( sin ) is the angular frequency (  ,   ,   are the velocity vector components).We neglect the inertial  2  term since its magnitude relative to the other terms is of order M 2  ≈ ( cool / ff ) −2 .We similarly neglect the  2  and     terms, since motion in the  direction is a result of the combination of radial and rotational motions, and hence   is of the same order as   or smaller.The momentum equation in the  direction is which indicates that the specific angular momentum  = Ω 2 sin 2  is conserved along flowlines, as expected under our assumption of axisymmetry.The mass and entropy equations (1) and (3) reduce to At large radii where the centrifugal terms in eqns.( 18)-( 19) are small, the solution will approach the spherical no-angular momentum solution discussed in section 2.1.In this limit we also expect   → 0, so eqn.(20) implies that angular momentum is independent of radius, i.e., it preserves the relation between  and  that exists in the outer boundary of the flow at  ∼  cool .We denote this boundary condition as where  c,max ≡  c ( = /2) and thus  c ( c,max ) c,max is the specific angular momentum at  = /2 (see eqn. 13), while  is some function that satisfies  (/2) = 1.The subscript '1' denotes that this relation is valid at large radii where   → 0. Equivalently, the angular frequency at large radii is Eqn. ( 23) implies that the circularization radius of a flowline which originates at polar angle  is ≈  c,max sin 2  (), with this expression being exact for constant  c .We show below that flowlines accrete onto the disk at a cylindrical radius equal to their circularization radius.To avoid flowline intersection, sin 2  () is assumed to monotonically increase at 0 ≤  ≤ /2, and thus the midplane flowline has the largest circularization radius.
The function  can be estimated using the results of non-radiative cosmological simulations, which provide the initial conditions in the hot gas before a radiatively driven inflow develops.Sharma et al. (2012a) found that Ω is weakly dependent on  in a sample of 10 11 −10 13 M ⊙ halos in such simulations, with differences of ≈ 15% between the midplane and the rotation axis.This suggests  () ≈ 1, so we use  () = 1 as our fiducial value.
We note that the Sharma et al. (2012a) simulations also suggest  ∝  0.5 −  0.7 at 0.1 < / vir < 1 (see their fig.2), i.e., the initial specific angular momentum profile of the hot gas increases outwards.We thus expect this increasing  profile to be replaced by a flat  ∝  0 profile (eqn.23) once the inflow develops.Assuming the outer boundary of the inflow expands with time as  cool grows (see section 2.1), then we expect the normalization of the  profile and hence  c,max to increase with time, as the inflow originates from larger radii where  is larger.This expected evolution however occurs on a cosmological timescale, and thus does not invalidate our steady-state assumption in the inner CGM which is achieved on the shorter cooling timescale at small CGM radii.Thus, for the purpose of deriving this steady-state solution in the inner CGM we treat  c,max as a constant.

Analytic solution in the slow rotation limit
In this section we derive a solution to equations (18)-( 22) which is accurate to lowest order in the effects of rotation.A similar approach was employed to study meridional flows in the Sun (Sweet 1950;Tassoul 2007) and in Bondi flows (Cassen & Pettibone 1976).
The dynamical effects of rotation on hot CGM inflows increase with decreasing , and become dominant at  ≲  c,max .To find the solution in the slow rotation limit we thus keep only terms which depend on  c,max / to the lowest order.It is straightforward to show that there are no terms of order (/ c,max ) −1 , since the lowest order of Ω is proportional to (/ c,max ) −2 (eqn.24) and rotation enters the other flow equations only via the term Ω 2  2 (eqns.18-19).We thus define a perturbation parameter and search for a solution of the form ,1 (, ) =   ,0 () ()    () Here, a subscript '0' denotes the non-rotating solution (eqn.10), a subscript '1' denotes the approximate solution which we wish to find, and   ,   ,    ,    are some functions of .The motivation for the form of   ,1 will become apparent below.The solution for Ω 1 is equivalent to eqn.(24) assuming  () = 1 as suggested by non-radiative cosmological simulations.The implications of other forms of  () on the solution are discussed below.We emphasize that the assumption of mild rotation is on top of the assumption of quasi-hydrostatic conditions, which allowed neglecting the quadratic velocity terms.Together, these assumptions imply that we assume the following conditions on timescales in the system: where  sc is the sound crossing time which is approximately equal to  ff since the flow is quasi-hydrostatic, and  rot = Ω −1 is the rotation time.
where  0 can be derived from  = (3/5)  2 s and eqn.(10): We thus get where C is a constant of integration to be determined below.Next, the first order terms in eqn.(18) give Using eqns.( 10), (25), and (29) then gives We can also define  1 =  0 (1 +    ).Since  ∝ / we get to first order in  that and similarly defining  1 =  0 (1 +    ) and using  ∝ / 5/3 we get Eqns. ( 30), ( 32) and ( 33) indicate that the pressure and density increase when traversing from the rotating axis to the midplane at a fixed , while the temperature decreases.The increase in pressure in the midplane is due to the higher density, which overcomes the lower effective gravity in the midplane which tends to decrease the pressure.
In the entropy equation ( 22), the second term on the left hand side is of order  2 (see eqns.26) and can be neglected.The first order terms of this equation are hence where we use  cool ∝ /Λ, and approximate the temperature dependence of the cooling function as a power-law Λ = Λ( 0 , )(/ 0 ) − .Further using  0 ∝  0 / 2/3 0 ∝  and  cool,0 = −/  ,0 based on eqn.(10), and the above relations for   ,   ,   , we get C .
(36) In the approximation on the right we use  = 0.5, appropriate for gas with  ∼ 10 6 K and a characteristic CGM metallicity of  ≈ 0.3 ⊙ (Miller & Bregman 2015).Last, we use the continuity equation ( 21) to derive   , which we cast in the form   ,1 =   ,0     (see eqn. 26).Keeping only first order terms we get Using the definition of  and that  0   ,0  2 is independent of , we then get so where D is another constant of integration.We further require   (/2) =   (0) = 0, in order to avoid a discontinuity at the rotation axis and to enforce symmetry with respect to the midplane.This gives D = 0 and C = −5/8, and hence Note that since   ,0 is negative, then   ,1 =   ,0     is positive for  < /2 and negative for  > /2, indicating that rotation diverts the flow towards the disc plane, as expected.
To summarize our solution we use the derived C = −5/8 in eqns.( 30), ( 32), ( 33) and ( 36) and get 2 sin( 2𝜃) where the zero-order terms are given by eqn.( 10).For a given  c , the solution in eqn.(41) depends on three parameters:  and Λ( 0 , ) (or equivalently CGM mass and metallicity) which set the non-rotating solution, and  c,max which sets the corrections due to rotation.The solution in eqn.( 41) is for an outer boundary condition in which Ω 1 is independent of  (i.e.,  () = 1).In Appendix A we give several solutions for Ω 1 () ∝ sin  () with integer , i.e., the rotation frequency at the outer boundary increases with angle from the rotation axis.In these solutions, the -dependent term in the solution for  1 is multiplied by a factor of sin 2 ()/( + 1) relative to that in eqn.(41), while the corresponding term for  1 is multiplied by a factor of sin 2 () • ( + 2)/( + 1).Thus, the result above that  and  increase towards the midplane, while  decreases, holds also when Ω 1 increases with .These deviations from spherical symmetry however tend to become weaker, and more concentrated near the midplane, with increasing .

Number of revolutions in CGM inflows
The number of revolutions around the rotation axis completed by a flowline can be derived from the ratio   /  implied by eqn.( 41): Using   ,0 = / cool (1 + O ()) and  ff = √ 2/ c we thus get It is thus evident that in solutions with larger  cool / ff the flowlines are more tightly wound, i.e. the flow rotates more prior to accreting onto the galaxy.The total number of radians a flowline rotates can be approximated with the following integral where in the first approximation we used eqn.( 43) and in the second approximation we used  cool / ff ∝  1/2 (eqn.10) and integrated over the range / c,max = 1 − 7. The upper bound of this range corresponds to the cooling radius (see section 2.1), though since the integrand scales as  −3/2 most of the rotation happens near  c,max , and the choice of upper limit does not significantly affect the result.Also, the lower bound of this range implies that neglecting the  () term is not formally justified.Below we validate this approximation with a more accurate numerical calculation.Equation ( 44) suggests that the number of rotations is set by  cool / ff near  c,max , where  cool / ff is estimated by the non-rotating solution (eqn.10).This can be understood intuitively since the cooling time tracks the inflow time ( cool = /|  |) while the free fall time tracks the rotation time near  c,max ( ff ≈ / c = /  ( c,max )).For the MW halo in which  cool / ff ( c,max ) ≈ 6 (eqn.16), we get that an accreting element rotates ≈ 12 radians prior to accretion.Furthermore, since  cool / ff increases with  c (see eqn. 10) we generally expect the amount of rotation to increase with galaxy mass.
It is informative to extend the result in eqn.( 44) also to galaxies with halo masses ≪ 10 12 M ⊙ , where  cool <  ff and the volumefilling phase is cool and in free-fall with   ≈ − c (e.g., Stern et al. 2020).Using −  = min(/ cool ,  c ) in eqn.( 44) we thus get Equation ( 45) demonstrates that only in the hot accretion mode where pressure-support slows down accretion relative to free-fall, the CGM has time to rotate significantly before accreting.In contrast, freefalling cold flows rotate merely by ≈ 1 radian prior to accretion.
The ratio  cool / ff at  =  c,max fully determines the hot rotating inflow solution, up to a scaling of the physical dimensions, as we show in Appendix B using non-dimensional analysis of the flow equations.We similarly show that the solution is fully determined up to a scaling by the ratio /  crit , where  crit is the critical accretion rate in which  cool / ff = 1 at  =  c,max (Stern et al. 2020).

Disk Accretion Radius
The distribution of disk radii at which gas accretes is an important input parameter for chemical evolution models and dynamical models of galaxy disks (e.g., Schönrich & Binney 2009;Krumholz et al. 2018;Wang & Lilly 2022).In this section we derive this distribution for hot and rotating CGM inflows.
Since the flow is spherical at large radii it holds that where we use  0 to denote the polar angle of a flowline at large radii.In hot rotating CGM inflows flowlines both conserve angular momentum (eqn.20), and accrete onto the disk at their circularization radius  c ( 0 ) (see numerical calculation below).The value of  c can be estimated from eqn. ( 23), which for constant  c and  ( 0 ) = 1 implies We thus get where the factor of 2 before the integral accounts for the two sides of the disk.Eqn. ( 48) indicates that the median accretion radius is (3/4) c,max , or 11.25 kpc for  c,max = 15 kpc, suggesting accretion weighted towards large disk radii.A similar conclusion of accretion mainly from large disk radii was deduced in FIRE cosmological simulations of Milky-Way like galaxies, in which the accretion is also dominated by hot rotating inflows (Trapp et al. 2022).

THE STRUCTURE OF HOT AND ROTATING CGM -NUMERICAL SOLUTION
In this section we derive numerical solutions for hot and rotating CGM inflows.To this end we run 3D hydrodynamic simulations until they converge onto a steady-state, where mass continuously flows through the hot CGM, cools and accretes onto the disk, and forms stars.Using this method to find the numerical solution has the advantage that it demonstrates that the solution is an attractor.The properties of the numerical solution are then compared to the approximate analytic solution derived in the previous section.

Setup
We use the meshless finite-mass ('MFM') mode of GIZMO (Hopkins 2015), a Lagrangian method with no inter-element mass flux, which enables us to track the history of each resolution element.The code accounts for self-gravity of the gas and stars, to which we add an acceleration term −( 2 c /) r with  c = 200 km s −1 .This term approximates the gravitational field in the inner halo due to unmodelled dark matter and stars.Optically thin radiative cooling is calculated using the  = 0 tables from Wiersma et al. (2009) down to  = 10 4 K, while optically thick radiative cooling to lower temperatures is disabled.All gas resolution elements with  SF > 10 cm −3 are converted into stellar particles.All stellar feedback processes are disabled.
The density, temperature and radial velocity of gas is initialized with a spherical, non-rotating hot inflow solution from Stern et al. (2019), to which we add rotation corresponding to some  c,max .This solution is found by integrating the 1D spherically-symmetric and steady-state flow equations, starting at  sonic = 0.1 kpc and proceeding outward.The integration uses the same  c = 200 km s −1 and cooling function with  = 0.3 Z ⊙ as in the simulation.The 1D solution has  = 1 M ⊙ yr −1 and at radii  ≫  sonic is well approximated by eqn.(10), with  = 2 • 10 6 K and Λ −22 = 0.3.We then randomly select initial positions in (, , ) for the initial location of gas resolution elements, such that the radial mass profile reproduces that in the spherically symmetric solution.To add a net rotation to the gas, all resolution elements at  >  c,max are initialized with   = 200 sin  (/ c,max ) −1 km s −1 , with  c,max = 15 kpc.This addition of rotation implies that the initial conditions are not in steady-state, since the initial pressure structure does not account for rotation support.We show below that the simulation adjusts to a new steady-state within a cooling time of < 1 Gyr, with a somewhat larger  of 1.7 M ⊙ yr −1 .
The mass of individual resolution elements is set to   = 1000 M ⊙ for elements at  < 100 kpc.This mass resolution implies a characteristic size of ≈ (3  /4 H ) 1/3 ≈ 0.2 −1/3 −3 kpc for typical hot gas densities of ≈ 10 −3  −3 cm −3 near the disk scale, and smaller sizes for the denser cool gas.For comparison, the height of the ≈ 10 4 K gaseous disk which forms from the cooling of the hot gas is ≈ 1 kpc2 .
Beyond 100 kpc the gas does not participate in the inflow since cooling times are too long, but needs to be included in the simulation in order to confine gas at smaller radii from expanding outward (in a realistic halo this confinement is achieved either by such hot gas with a long cooling time or by the ram pressure of infalling gas outside the accretion shock).To avoid investing too much computing time in this confining outer gas, we sample the spherically-symmetric solution beyond 100 kpc with resolution elements which masses increase by a factor of three every factor of √ 2 in radius, out to 3.2 Mpc where the sound-crossing time equals 10 Gyr.In total the CGM is simulated with 3.1 × 10 7 resolution elements.
We also add a galaxy to the initial conditions, using the MakeDisk code (Springel et al. 2005) with the following parameters.The stellar disk is initialized with mass  * = 10 8 M ⊙ , cylindrical radial scale length of  d =  c,max /4 = 3.75 kpc spanning 0.03 − 4 d , and a vertical scale length of 0.1 d .The gaseous disk has a mass  disk gas = 0.2 * , and the same exponential distribution as stars, and the bulge has a mass  bulge = 2 • 10 7 M ⊙ and scale length 0.1 kpc.We include in the MakeDisk calculation the same isothermal gravitational field used in the hydro simulation, and stellar and gas particles in the disk have the same   = 1000 M ⊙ resolution as in the CGM.The choice of galaxy parameters is inconsequential as long as the initial mass is small compared to the accreted mass, which at  = 1 Gyr is  ∼ 10 9 M ⊙ .The simulation is run for 3.5 Gyr, with snapshots saved every 5 Myr.At all times and radii the gravitational field is dominated by the included isothermal gravitational field with  c = 200 km s −1 , rather than by the simulated gas and stars.
Our setup is loosely based on the setup of Su et al. (2019Su et al. ( , 2020)), which simulated the behavior of gas in group and cluster-sized halos.A similar setup to ours for Milky-Way mass halos was employed by Kaufmann et al. (2006) using an SPH code.This code was later found to over-predict artificial clumping of the cool gas (Agertz et al. 2007;Kaufmann et al. 2009).Our use of the MFM code addresses this numerical issue (see Hopkins 2015 for code tests).Additionally, since in our simulation all the hot gas cools once it inflows past  c,max (see below), the numerical details may affect the distribution of clumps and their typical sizes, but not the total mass which cools.

Overview of results
Figure 1 shows temperature maps in the simulation at  > 1 Gyr, after the hot CGM phase converged onto an axisymmetric steady-state solution within  ≈ 40 kpc.Steady-state and axisymmetry are evident from the small dispersion in hot CGM properties with time and , as shown below.The left and middle panels respectively show the  −  plane and the  −  plane (mass-weighted over −10 <  < 10 kpc).The figure shows that the hot gas fills the volume except in the disk region at  ≲  c,max = 15 kpc and || ≲ 1 kpc.Black lines depict flowlines in the two planes, derived as described below.Three of these flowlines are also depicted in the right panel as 3D 'tubes', rendered using the Mayavi software (Ramachandran & Varoquaux 2011) on snapshot data.The tube cross-section scales as () −1 and thus illustrates the compression of the flow.
Figure 1 shows that the flowlines in the hot gas are helical, with the hot gas spiraling onto the galaxy.While inflowing, the gas initially remains hot with  ∼  vir , and then cools to ∼ 10 4 K just prior to joining the ISM disk.This cooling is accompanied with strong compression of the flow, as evident from the sharp decrease in the width of the flow tubes in the right panel (tube thickness should drop to ≲ 0.01 pixels upon cooling according to the () −1 scaling, though is plotted with one pixel for visibility).
Figure 2 plots radial shell-averaged velocities in the simulation after steady-state is achieved.The top panel shows that at radii  >  c,max the sound speed  s (red) approximately equals  c (gray), indicating the hot gas is to first-order supported against gravity by thermal pressure, as indicated also by the slow inflow velocities of −  ≪  c (magenta line).At radii  >  c,max the rotation velocity increases inward roughly as  −1 due to conservation of angular momentum, reaching   =  c at  c,max .Within  c,max the gas is sp he ri ca l di st ri bu ti on fully rotationally supported and cool with   ≪  c , and the radial velocity drops to zero.The associated change in geometry is evident in the bottom panel, which plots the average absolute height above the midplane || in different radial shells.The gas distribution is close to spherically-symmetric at  >  c,max , in contrast with a thin disk distribution at  <  c,max .

Accretion of the hot CGM onto the cool ISM
Figure 3 provides a Lagrangian view of hot inflowing CGM, by plotting median properties of resolution elements versus time since the element was at  = 40 kpc.The properties depend on the initial polar angle of the flowline  0 , so for each  0 in (0.1, 0.3, 0.4, 0.5) we group all resolution elements that reside at 40 <  < 41 kpc and | −  0 | < 0.025 at times 1 <  < 1.5 Gyr.Then, for each  0 group we plot the median and 16 − 84 percentile ranges of , , , and   .The 16 − 84 percentile range thus accounts for the dispersion both with  and with , and specifically a small 16 − 84 percentile range indicates that the solution is axisymmetric and in steady-state.Fig. 3 shows that at  ≳ 25 kpc (early times in this plot), the gas is hot and inflowing, with a somewhat larger inflow velocity in the  0 = 0.1 flowline near the rotation axis.Rotation is sub-Keplerian (  <  c ≈ 200 km s −1 ) but growing with time, as also indicated by Fig. 2. The value of  remains roughly constant and equal to  0 for a given flowline.Then, when the flowlines reach radii of  ≲ 20 kpc, the gas initially heats up and is diverted to the midplane ( = /2), and then abruptly cools.The initial heating is more pronounced near vidual flowlines are small, indicating the hot CGM is axisymmetric and in steady-state.Furthermore, cooling and accretion happens to all hot inflow resolution elements, indicating that cooling is a global transition in the inflow, rather than occuring only in individual gas clumps as in precipitation models (e.g.Maller & Bullock 2004;Voit et al. 2017).
Figure 4 plots gas properties along flowlines versus  −  (10 5 K), where  (10 5 K) is defined as the time at which the temperature in the flowline equals 10 5 K.This time is also marked with vertical lines in Fig. 3, and as mentioned above is equivalent to the time of accretion onto the ISM.The twelve panels show cylindrical radius , , total rotation  −  0 where  0 ≡ ( = 40 kpc), rotation velocity, density, temperature, pressure, entropy, density dispersion, specific angular momentum, radiative cooling rate per unit mass  2 H Λ/, and compressive heating rate per unit mass / where  =  −1 is the specific volume.The figure shows that while the flow is hot (at  <  (10 5 K)), the density and pressure of the hot inflow mildly increase with time, while the temperature is roughly constant and the entropy decreases.The increase in density is due to the contraction of the inflow, which also causes a compressive heating ate of order / ≈ 0.005 erg s −1 g −1 .This compressive heating offsets comparable radiative losses (compare panels k and l), thus keeping the temperature roughly constant while the entropy decreases.Also, panel i shows that density fluctuations are small (⟨/ ρ⟩ ≪ 1) as in a non-rotating cooling flow (Balbus & Soker 1989;Stern et al. 2019), and panel j shows that the specific angular momentum is conserved since the system is axisymmetric.Panel c shows that flowlines rotate 4 − 8 radians prior to cooling, or roughly one full revolution.
Fig. 4f shows that at  ≈  (10 5 K) when the flow abruptly cools, the density increases by a factor of ≈300 for the flowline in the midplane ( 0 = 0.5), and by a larger factor in flowlines with smaller  0 .At the same time   slightly increases (panel j), likely as a result of torques by stars and preexisting disk gas.Also apparent is that density fluctuations become strong just before the gas cools (at  −  (10 5 K) ≈ −25 Myr), and remain of order unity after cooling, in contrast with the weak density fluctuations when the flow is hot (panel i).The transition to a disk geometry starts somewhat earlier, at  −  (10 5 K) ≈ −250 Myr (panel b, and also  panel in Fig. 3).
The eventual drop in temperature from  ≈ 2 • 10 6 K to  ≈ 10 4 K at  (10 5 K) is an inevitable result of the inflow halting due to rotation support, which stops compressive heating.Absent any heating sources, the gas cools on a cooling timescale, which is ∼ 10 Myr at  (10 5 K).This short cooling timescale is a result of the flattening to a disk geometry which increases the density to  H ≳ 0.01 cm −3 (see zoom-in on  ≈  (10 5 K) in Figure C1).This layer where the hot inflow cools corresponds to the disc-halo interface (e.g., Fraternali & Binney 2008;Marasco et al. 2012;Fraternali 2017), also known as 'extraplanar gas', which is further addressed in the discussion.
The result that density fluctuations remain small when the flow is hot (Fig. 4i) is potentially due to the accretion process occurring on a timescale comparable or shorter than  cool on which thermal perturbations develop.For example, at 0.5 Gyr prior to accretion (i.e., at  −  (10 5 K) = −0.5 Gyr) we find  cool = 1.7, 2.9, 5.3 and 12 Gyr in the four flowlines shown in Fig. 4, with longer  cool closer to the rotation axis due to the higher temperature and lower densities.The 0.5 Gyr remaining until accretion is thus insufficient for the thermal instability to grow, despite that the hot gas is formally unstable.A similar argument explains why significant density perturbations do not develop spontaneously in non-rotating cooling flows (Balbus & Soker 1989).This conclusion is also consistent with the simulations in Sormani & Sobacchi (2019), which found that condensations develop only when a heating term is added to part of the hot gas, thus increasing the accretion time relative to a cooling flow.

Deviations from spherical symmetry in hot inflowing CGM
Rotational support in the hot CGM is expected to induce lower gas densities and higher gas temperatures along the rotation axis relative to the midplane (Barnabè et al. 2006;Pezzulli et al. 2017;Sormani & Sobacchi 2019).In this section we quantify these deviations from  -4).Magenta lines plot the analytic non-rotating solution (eqn.10), while blue lines plot the analytic slow-rotating solution (eqn.41) which accounts only for lowest-order terms in  = (/ c,max ) −2 (noted on top).In the rotating solutions density and pressure increase towards the midplane, while temperature decreases.
spherical symmetry in our simulation of a hot rotating inflow, and compare it to the analytic approximation deduced above (eqn.41).
Figure 5 plots the dependence of hot CGM properties on polar angle  at radii of 45 kpc and 25 kpc.From top to bottom the different rows plot angular frequency, temperature, hydrogen density and thermal pressure.Magenta lines denote the non-rotating analytic solution (eqn.10), blue lines denote the slow rotating analytic solution (eqn.41), and black lines the solution in the simulation after steady-state is achieved (the same solution used in Figs.1-4).The perturbation parameter  = (/ c,max ) −2 defined in eqn.( 25) is noted at the top.The slow-rotating analytic solution accounts only for the lowest-order terms in this quantity.
Fig. 5 demonstrates how the properties of the hot gas deviate from spherical symmetry due to the rotation, and more so at radii approaching  c,max where rotation support is more significant.At  = 25 kpc in the numerical solution, the temperature at the rotating axis is almost a factor of two lower than in the midplane, while the Table 1.Parameters of simulations used in Figure 6.
[km s  a) Derived from the four other parameters using eqn.( 10). (b) Fiducial simulation used also in other figures.
density is a factor of two higher.Note also that the slow-rotating analytic solution rather accurately reproduces the simulation at  = 45 kpc where  = 0.1.At  = 25 kpc where  = 0.4 the analytic solution is qualitatively consistent with the trends of ,  H , and  versus , though there are quantitative differences potentially since high-order terms in  are neglected in the analytic solution.

Revolutions in inflow versus 𝑡 cool /𝑡 ff
To test the relation between total rotation in the inflow and  cool / ff measured at  c,max (section 2.5), we run several simulations with different combinations of ,  CGM , and  c which yield different  cool / ff via eqn.( 10).The parameters of the simulations are listed in Table 1.For  we use the value measured through a shell at 2 c,max in snapshots after the simulations achieve steady-state, which is typically 10−75% larger than  in the initial conditions (see §3.1).
On these snapshots we also measure the average rotation Δ = ∫ Ω a fluid element completes as it inflows from 10 c,max to  c,max , and plot them in Figure 6.The figure shows that the simulations roughy follow the analytic estimate from eqn. ( 45), confirming that the number of rotations in hot rotating CGM scales with  cool / ff .

Disk accretion radius
Figure 7 plots the accretion rate onto the disk within cylindrical disk radius .Solid line is based on the fiducial simulation, which we measure by calculating the mass flux through two disk-shaped surfaces with radius  located at  = ±1 kpc, in a snapshot after the simulation achieved steady-state.The step at  =  c,max = 15 kpc corresponds to planar accretion from the disk edge, derived from the mass flux through a closing vertical surface spanning −1 <  < 1 kpc at  =  c,max .The simulation result is close to the analytic estimate in eqn.( 48), which is based on the assumption that each flowline in the hot inflow accretes at its circularization radius.

ADDITIONAL CONSIDERATIONS
In this section we consider the effect of additional physical mechanisms and effects which were not included in the simulation above.

Viscosity
Viscous forces in the flow may in principle cause angular momentum transport along directions where there is shear in the flow.In our solution Ω ∝  −2 up to corrections of order  (eqn.41) and thus there is shear in the radial direction.Here we show that for standard kinematic viscosity the expected angular momentum transport due to this shear is small.The specific angular momentum of the flow is  = Ω 2 sin 2  =  c  c,max sin 2  (eqn.41).Viscous forces in the radial direction will cause an angular momentum loss per unit time of where  is the kinematic viscosity (e.g., Fabian et al. 2005) and   is the reduction of the viscosity relative to the Spitzer value.We can estimate the fractional angular momentum loss due to viscous forces by multiplying / by the flow time ≈  cool and dividing by : where we used the analytic solution in eqn.( 41) and neglected corrections of order ( c,max /) 2 .For typical estimates of   ∼ 0.1 (Narayan & Medvedev 2001) this value is substantially smaller than unity, indicating that viscous forces can generally be neglected.

Turbulence
Assuming that turbulence is seeded at large CGM radii, for example by cosmological accretion or due to stirring by subhalos, what would be the fate of these turbulent motions in the inner CGM inflow The dashed line is the analytic estimate, which assumes each flowline accretes at its circularization radius (eqn.48), and provides a good match to the simulation result.This predicted  disk profile in hot rotating inflows can be useful for chemical evolution models and dynamical models of galaxy disks (e.g., Schönrich & Binney 2009;Krumholz et al. 2018).
explored here?In a non-rotating inflow, we expect a balance between dissipation of turbulence on a timescale  diss = / turb and 'adiabatic heating' of turbulence due to the contraction of the inflow on a timescale  flow = /  (Robertson & Goldreich 2012).This balance suggests that contracting turbulent fluids converge to  flow ∼  diss and hence  turb ∼   , since more rapid turbulent motions will dissipate while slower turbulence will heat up (Murray & Chang 2015;Murray et al. 2017).In a steady-state cooling flow where  flow ≈  cool , we thus expect also  diss ∼  cool .Using  ff ∼ / c and  c ≈  s , it thus follows that Since  ff <  cool , eqn. ( 52) suggests that turbulence is subsonic, i.e., turbulent support is subdominant to thermal support, as assumed in section 2.1.Furthermore, this relation suggests that relative importance of turbulent motions decreases with increasing  cool / ff .Note that eqn.( 52) is based on the assumption that the dominant turbulence driving mechanism at inner CGM radii is adiabatic 'heating' of pre-existing turbulence in the inflow.This assumption is similar to the underlying assumption of our solution that the dominant thermal heating mechanism is compression of the CGM inflow, rather than other heating sources such as feedback.
In a rotating inflow, turbulence may also be induced by the shear between adjacent shells.Radial displacements due to such turbulent motions are not subject to restoring Coriolis forces, since the nonperturbed solution is angular momentum conserving (Fig. 4) and hence the epicyclic frequency  is zero.Balbus et al. (1996) showed that for a rotationally-dominated disk with  = 0 such turbulence develops with an e-folding time roughly equal to an orbit time.Given that hot CGM inflows onto Milky-Way like galaxies complete ≲ 2 orbits before accretion (Fig. 6 with  cool / ff ≈ 10) it is unclear if shear-induced turbulence has sufficient time to develop.As a preliminary test of the effect of turbulence on our hot rotating CGM solution, we run another simulation with similar initial conditions as in our fiducial simulation, to which we add a turbulent velocity field with amplitude  turb ( = 0) = 30 km s −1 and a lognormal power spectrum peaking at a wavelength of 50 kpc and a logarithmic width of 1. Figure 8 compares the results of this simulation after steady-state is achieved with the results of the fiducial simulation shown in Figs.1-5.The top panel shows that adding turbulence increases the density fluctuations in the hot inflow, though they are still below unity at  <  (10 5 K).The bottom panel demonstrates the effect of turbulence on the specific angular momentum in the flow.The turbulent simulation shows a smaller difference in   between different flowlines than in the fiducial simulation, likely as a result of angular momentum transfer between flowlines.Also evident is the mild (≲ 20%) decrease in   in the turbulent simulation over the last 1.5 Gyr prior to cooling, in contrast with the constant   in the fiducial simulation.This is potentially a result of transport of angular momentum outward during the inflow due to the turbulent viscosity between adjacent shells.Fig. 8 thus implies that turbulence, at the strength explored, has only a mild effect on the angular momentum content of the inflow, and specifically ≳ 80% of the angular momentum in the inflowing hot CGM accretes onto the ISM3 .
The development of turbulence in hot and rotating CGM inflows also affects the angular momentum distribution of accreting gas, since a large  turb necessary implies a broad angular momentum distribution.Hafen et al. (2022) argued that a narrow angular momentum distribution in accreting gas may be necessary for the formation of thin disk galaxies seen in the low-redshift Universe.This points to the importance of understanding turbulence in hot rotating CGM.We defer a detailed analysis to future work.

Magnetic fields
The contraction and rotation in hot rotating CGM inflows is expected to enhance magnetic fields present at the outer radius of the inflow.In appendix D we estimate this enhancement using the rotating inflow solution derived above, assuming ideal MHD and ignoring potential dynamical effects of the magnetic field on the flow.Defining  0 as the outer radius of the inflow, and assuming an isotropic seed field   ( 0 ) =   ( 0 ) =   ( 0 ) ≡  0 , we find that where we also defined  0 ≡   ( 0 ).The values of   / 0 ,   / 0 , and   / 0 are accurate up to corrections of order ( c,max /) 2 .The enhancement of the magnetic field in eqn.( 53) can be compared to the enhancement in a non-rotating spherical inflow, which was derived by Shapiro (1973).They found   ∝  −2 and   ∝   ∝ (  ) −1 , which can be understood intuitively as a result of conservation of magnetic flux through patches moving with the flow.Rotation thus mainly affects the enhancement of   , due to the winding of the field.Eqn.(53) shows that the enhancement of   is a product of ( cool / ff •  c,max /) and ( 0 /) 2 , where the former tracks the number of radians rotated by the inflow (eqn.44) and the latter tracks the contraction of an inflowing shell.
For  0 / c,max ∼ 6 and  cool / ff ∼ 6 as expected in the hot Milky-Way CGM (section 2), eqn.( 53) suggests an increase in   of order ∼ 200 by the time the hot gas reaches  c,max , just prior to accreting onto the galaxy.For comparison, the thermal pressure increases over the same range of radii as ( c,max )/( 0 ) ≈ ( c,max / 0 ) −3/2 ∼ 15 (eqn.10), and thus the ratio of thermal to magnetic pressure  ∝ / 2 is expected to decrease by a large factor of ∼ 3000.Current upper limits on the magnetic field in the inner CGM of ∼  ★ galaxies at  ≲ 0.3 suggest magnetic pressure is subdominant to the thermal pressure ( > 1), at least along the major axis where most of the accretion is expected (Prochaska et al. 2019;Lan & Prochaska 2020;Heesen et al. 2023).It thus follows that if the hot CGM is accreting as suggested in this work, seed magnetic fields are sufficiently small that they do not dominate even after the large enhancement induced by contraction and rotation.
The eventual cooling of the inner hot CGM onto the ISM will further enhance .Fig. 4f suggests that the gas density increases by a factor of ∼ 1000 as it cools, which would increase  by a further in the fiducial simulation.This finite  is potentially sufficient to prevent any further development of hydrodynamic turbulence due to restoring Coriolis forces (Balbus et al. 1996).
factor ∼ 1000 2/3 = 100 in the limit of ideal MHD.Another potentially interesting implication of the hot CGM solution concerns the development of turbulence due to the magnetic-rotational instability (MRI).The MRI amplitude growth rate is ∼ Ω (e.g.Balbus & Hawley 1998;Masada & Sano 2008), so the result that ∫ Ω ≈  cool / ff (Fig. 6, eqn.44) implies that prior to accretion MRI can grow by   cool / ff , i.e. a factor of 10 4 for  cool / ff ≈ 10.The solution may thus change considerably as  cool / ff exceeds some critical value where MRI becomes fully developed.We defer analysis of accretion via magnetic hot rotating CGM inflows to future work.

OBSERVATIONAL IMPLICATIONS
In this section we discuss several observational signatures of hot and rotating CGM inflows.While it would be challenging to detect the predicted slow radial velocities of a few tens of km s −1 (eqn.10), the predicted rotation velocities are faster, reaching  c ∼ 200 km s −1 , and indeed evidence for such fast hot gas rotation has been detected in the Milky-Way CGM (Hodges-Kluck et al. 2016).Since an inflow imparts a specific rotation profile in the hot gas which is flatter (  ∼  −1 , eqn.23) than the rotation profile expected prior to the development of an inflow (  ∼  −0.5 , Sharma et al. 2012b;Pezzulli et al. 2017;Sormani et al. 2018), measuring the rotation profile could be used to support or rule out the existence of inflows in the hot gas.
The hot gas rotation pattern can potentially be detected directly using X-ray emission line centroiding (5.1), or indirectly, by identifying the lower densities and higher temperatures along the rotation axis relative to the midplane (5.2-5.3).The latter indirect signature would however have to be distinguished from qualitatively similar trends induced by feedback (Truong et al. 2021;Truong et al. 2023;Nica et al. 2021;Yang et al. 2023).
Given the idealized nature of the solution, signal strength estimates below are at the order of magnitude level.More realistic calculations based on cosmological simulations would be a useful next step.-Kluck et al. (2016) analyzed the centroids of 37 Ovii absorption lines in the Milky-Way CGM, in the context of a phenomenological hot CGM model with  H ∝  −1.5 and uniform inflow and rotational velocities.They deduced   = −15 ± 20 km s −1 and   = 180±41 km s −1 , or   / c = −0.06±0.08 and   = 0.75±0.17for a circular velocity of  c = 240 km s −1 measured near the Sun.These values are comparable to those expected in hot rotating inflows near the disk, since   / c ≈  c,max / and   / c ≈  ff / cool (Fig. 2, eqns.41 and 10).Specifically, we calculate the Ovii absorptionaveraged   and   in our fiducial simulation, along sightlines starting at (, ) = (8 kpc, 0 kpc) with the same distribution of Galactic latitudes and longtitudes as in the Hodges-Kluck et al. sample.We find a median   / c = 0.62 and   / c = −0.1,consistent with their results.The Hodges-Kluck et al. observations of hot CGM kinematics in the Milky-Way halo are thus consistent with a rotating inflow.

Hodges
A more accurate test of the predicted hot CGM rotational velocity profile can be performed with an X-ray microcalorimeter with high spectral resolution.Top panel of Fig. 9 et al. 2004).The projection is done on a snapshot after the simulation converged onto a steady-state.Line emissivity is calculated based on the gas density and temperature in the simulation using the pyXSIM package (ZuHone & Hallman 2016)4 .Pixel size in this panel is 3 kpc, corresponding to the planned 15 ′′ resolution of the proposed Line Emission Mapper probe (LEM, Kraft et al. 2022) for a target at a distance of 40 Mpc5 .The bottom panel shows the line centroid difference between the approaching and receding sides of the CGM, in the midplane.This difference reaches 200 km s −1 near the disk, higher than the planned centroiding accuracy of ≲ 70 km s −1 planned for LEM.X-ray telescopes with high spectral resolution may thus be able to measure the rotation velocity profile in the hot CGM, and test whether it is consistent with the inflow solution.Oviii Ly/Ovii He emission line ratio.The ratio increases with angle due to higher temperatures near the rotation axis (see Fig. 5).

Angle dependence of X-ray emission and temperature
Angular momentum support induces deviations from spherical symmetry in the hot CGM density and temperature (section 3.4, Fig. 5).These are potentially detectable by measuring the dependence of CGM X-ray emission on the 'azimuthal angle', defined as the orientation of the sightline with respect to the galaxy major axis (e.g., Kacprzak et al. 2015).The top panel in Figure 10 shows the predicted soft X-ray surface brightness versus azimuthal angle and impact parameter  ⊥ , assuming CGM rotation is oriented edge-on in the plane of the sky.Surface brightness is calculated using pyXSIM on the NGC 891-like simulation used also in Fig. 9.The figure shows that the soft X-ray brightness (∝  2 H ) increases towards the major axis at small  ⊥ , since rotation induces a higher CGM density near the midplane (eqn.41, Fig. 5).
The bottom panel in Fig. 10 shows the luminosity ratio of the Oviii Ly 0.65 keV and Ovii He 0.56 keV emission lines.The ratio increases towards the minor axis, by up to 75% at  ⊥ = 30 kpc.This follows since this emission line ratio depends mainly on CGM temperature which is higher along the rotation axis (see Fig. 5).

Dispersion Measure
Observations of the dispersion measures to pulsars in the Magellanic clouds and towards extragalactic Fast Radio Bursts (FRBs) constrain electron column densities in the CGM (e.g.Anderson & Bregman 2010;Prochaska & Zheng 2019;Williams et al. 2023;Ravi et al. 2023).Figure 11 plots predicted dispersion measures for an external galaxy and for the Milky-Way CGM, based on our fiducial simulation.
For external galaxy sightlines we assume an inclination  = 77 • similar to M31, and integrate the electron density along sightlines with different impact parameters  ⊥ and different azimuthal angles.We start and end the integration at a spherical radius  = 100 kpc, roughly equal to  cool (section 2.1) beyond which the hot inflow solution does not apply.Fig. 11 shows that the highest dispersion measures of ≈ 30 cm −3 pc are found at small impact parameters and small azimuthal angles where densities are highest (see eqn. 41).
For Milky-Way sightlines the dispersion measures were calculated by integrating from (, ) = (8 kpc, 1 kpc) out to  = 100 kpc, for different Galactic latitudes  and Galactic longtitudes .We start at  = 1 kpc to avoid the contribution of the cool disk, while the exact choice of outer limit does not significantly affect the result since most of the contribution comes from small radii.The predicted dispersion measures are 12 − 18 cm −3 pc and increase towards lower , again due to higher densities near the disk plane.Fig. 11 also shows the upper limit of 23 cm −3 pc for the CGM dispersion measure from Anderson & Bregman (2010), deduced based on sightlines to pulsars in the LMC (though note caveat in Ravi et al. 2023, which may imply this upper limit is too restrictive).This upper limit is consistent with our prediction of 14 cm −3 pc in sightlines with || = 30 • and  = 270 • when integrating out to an LMC distance of  = 50 kpc.

Comparison to previous hot CGM models
In the solution described in this work the hot CGM is inflowing, since ongoing feedback heating is assumed to be small relative to radiative losses.This is in contrast with the typical assumption employed by models of the low-redshift CGM, where the hot CGM phase is static due to feedback heating balancing radiative losses (e.g.Sharma et al. 2012b;Voit et al. 2017;Faerman et al. 2017Faerman et al. , 2020)).Several models have accounted also for rotation in the hot CGM (Pezzulli et al. 2017;Sormani et al. 2018;Afruni et al. 2022), though also in these latter models the gas is assumed not to flow in the radial direction.A third possibility is that feedback heating exceeds radiative losses, and the hot CGM forms an outflow (e.g., Thompson et al. 2016;Schneider et al. 2020, though note these studies neglected a pre-existing CGM).As the mechanics of CGM heating by stellar and AGN feedback are currently not well understood, and existing X-ray constraints do not rule out an inflow as in the ICM (see introduction), it is currently unclear which of these three paradigms is more accurate.
Despite the qualitative distinction between hot inflows explored here and static hot CGM models explored by previous studies, both types of models satisfy similar hydrostatic equilibrium constraints.This follows since the inflow solution is highly subsonic, and hence

Milky Way
Figure 11.Predicted dispersion measures from a CGM forming a hot rotating inflow, for sightlines through an external galaxy CGM (left, assuming M31-like inclination) and through the Milky-Way CGM (right), based on the fiducial simulation.External galaxy sightlines are given as a function of impact parameter and azimuthal angle, while Milky-Way sightlines as a function of Galactic coordinates.Integration is limited to  < 100 kpc where a hot inflow is possible.The upper limit from Anderson & Bregman (2010), derived from sightlines to pulsars in the LMC, is noted in the right panel.The decrease in dispersion measure with increasing azimuthal angle and  is due to the lower densities near the rotation axis.
deviations from hydrostatic equilibrium in the radial direction are small, of order ( cool / ff ) −2 .Indeed, these deviations are neglected in the analytic solution derived in section 2.4.The assumption of an inflow however enforces conservation of mass, energy and angular momentum between adjacent shells (eqns.20-22), while there are no similar constraints on static models.The allowed space of inflowing hot CGM solutions is thus significantly smaller than that of static solutions.For example, deducing the entropy profile in static models requires employing another assumption, such that the entropy is independent of radius (Faerman et al. 2020), or that  cool / ff is independent of radius (Voit et al. 2017).In the inflow solution derived here the entropy profile is fully determined by the flow equations.
Inflow solutions are also more specific than static models in the predicted rotation profile of the hot CGM.Sormani et al. (2018) used several constraints to derive   (, ) in their static hot CGM model, mainly the typical CGM angular momentum distribution in cosmological simulations and Ovii absorption-based estimates of Milky-Way CGM rotation from Hodges-Kluck et al. (2016).These constraints are however rather sparse and leave considerable freedom for different choices of   (, ), as discussed in Sormani et al.In contrast, in the inflow solution derived here angular momentum is conserved along flowlines (eqn.20), and thus to fully determine   (, ) one only has to specify the angular momentum of each flowline.In our fiducial model, the midplane flowline has an angular momentum corresponding to the typical spin of dark matter halos (section 2.2), while the angular momentum of other flowlines is set by requiring a constant angular frequency at the inflow outer boundary at  ∼  cool ∼ 100 kpc, as suggested by non-radiative cosmological simulations (section 2.3).The predicted   (, ) is consistent with the Milky-Way Ovii absorption measurements (section 5.1).Also, the inflow solution predicts a flat angular momentum profile at  c,max <  ≲  cool (eqn.23), as often seen in FIRE cosmological simulations of Milky-Way mass galaxies at  ∼ 0 (Stern et al. 2021a, see figure 13 there).

Accretion via hot inflows versus 'precipitation'
Accretion onto the Milky Way and nearby spirals is likely dominated either by 'precipitation', where local thermal instability in the hot phase creates ∼ 10 4 K gas clumps which lose buoyancy and accrete (e.g., Fall & Rees 1985;Maller & Bullock 2004;Voit et al. 2015;Armillotta et al. 2016), or by the hot ∼ 10 6 K inflows discussed in this work (see introduction).Both accretion modes would be considered 'hot accretion' in the context of the classic distinction between the hot and cold accretion modes, since they both originate in the hot phase of the CGM (e.g.Nelson et al. 2013).However, in the scenario studied here the CGM inflow remains at ∼ 10 6 K down to a cylindrical radius  c,max ∼ 15 kpc and height of ∼ kpc above the midplane, at which point all the hot gas cools and joins the ISM, rather than just a subset of localized clouds.Hot inflows are thus a type of 'quiet accretion' (Putman et al. 2012) -accretion which becomes accessible to cool gas observations only when it cannot be kinematically distinguished from pre-existing disk gas.
We note that at outer CGM radii of  ≳  cool ∼ 100 kpc a hot inflow is not expected since cooling times are long (section 2.1), so any infall would likely be dominated by cool ∼ 10 4 K gas.This cool inflow at large radii can potentially join a hot inflow at small CGM radii if the cool clouds are disrupted by hydrodynamic instabilities (e.g., Tan et al. 2023;Afruni et al. 2023).

Hot inflows require that CGM of typical ∼ 𝐿 * spirals have previously expanded due to feedback
In steady state we expect a CGM inflow rate of  ≈ 0.6 SFR, which is ≈ 1 M ⊙ yr −1 for the Milky-Way (section 2.1).We show here that if this  originates in a hot inflow, the CGM density must be lower than expected in a baryon-complete halo.
Integrating the density profile in eqn.(10) out to the virial radius and solving for  we get: There is however an inherent challenge in a scenario where discs are fed by an inflow from an expanded hot CGM.An expanding CGM is apparently contradictory to an inflowing CGM, and requires feedback heating to dominate radiative cooling, in contrast with the above assumption that feedback heating is subdominant.This apparent contradiction can be circumvented if the inflow and expansion are separated either in time or in space.For example, it is plausible that feedback was strong high redshift causing the CGM to expand, and has since subsided so the hot CGM develops an inflow at low redshift.Such an evolution in feedback strength is predicted by FIRE simulations of Milky-Way mass galaxies (Muratov et al. 2015;Faucher-Giguère 2018;Stern et al. 2021a;Pandya et al. 2021), and is also consistent with observed stellar winds which are strong in  ∼ 2 galaxies but weak in typical ∼ ★ galaxies at  ∼ 0 (Heckman & Thompson 2017).Alternatively, black hole feedback or stellar feedback which occur in 'bursts' separated by more than ∼ 1 Gyr would allow a hot inflow to develop between bursts, since this timescale is the cooling time in the inner CGM on which a hot inflow develops.A third possibility is that feedback mainly heats the outer CGM, allowing the inner CGM to form an inflow.While this may seem counter-intuitive, it is potentially possible if feedback is focused on the rotation axis at small CGM radii and isotropizes only at large CGM radii, thus allowing the hot inner CGM to inflow from the midplane.Another option is that feedback energy is propagated by weak shocks or sound waves which dissipate and dump heat only in the outer CGM.The distribution of feedback energy in space and time has been explored extensively in the context of the ICM (e.g., Yang & Reynolds 2016;Martizzi et al. 2019;Donahue & Voit 2022), but less so in the context of the CGM.A thorough exploration would allowing understanding under which conditions the hot CGM inflow scenario is viable.

Disk-halo interface
The cooling layer of the hot inflow solution found above, in which the gas temperature drops from ≈ 2 • 10 6 K to ≲ 10 4 K, occurs at || ≲ 0.5 kpc in our fiducial simulation (see Figs. 1 and C1).This simulated cool layer is thinner than the 'extraplanar' layers of neutral and ionized ∼ 10 4 K gas detected around nearby spirals, which extend to || ≈ several kpc (e.g., Sancisi et al. 2008;Gaensler et al. 2008).Also, the mass flow rate through this layer in our simulation is  ≈ 1 M ⊙ yr −1 ∼ SFR, in contrast with ∼ 10 × SFR in observed extraplanar gas layers (e.g., Marasco et al. 2019).
Extraplanar gas is often assumed to be dominated by fountain flows (Shapiro & Field 1976;Bregman 1980), in which cool clouds driven upward by feedback penetrate into the hot CGM and then fall back onto the disk (Fraternali & Binney 2006, 2008;Marasco et al. 2012;Fraternali 2017).Fountain flows thus both increase the vertical extent of the cool gas and the total mass circulating through the extraplanar gas layer.These models usually assume the hot CGM is not inflowing, which as mentioned above requires a delicate balance between feedback heating and cooling, a balance especially hard to achieve in the inner CGM where cooling times are short (see also discussion of this issue in Sormani & Sobacchi 2019).It would thus be beneficial to study how fountain flows interact with the inflowing hot CGM solution derived herein.Specifically, one could use the angular momentum structure of a hot inflow derived above to understand how fountain orbits are affected by angular momentum exchange with the hot CGM, an interaction that has observable implications (e.g., Fraternali 2017).Also it would be useful to understand how this fountain -hot CGM interaction affects the accretion rate profile onto the disk (compare Figure 7 above with figure 9 in Marasco et al. 2012).
We note that some of the above studies argued that fountain flows are required for accretion from the hot CGM, since they trigger cool cloud condensation in the hot gas.Our simulations show that the hot CGM cools and can fuel star formation even without fountain flows.
Cooling in our solution is however a steady process which affects the entire hot inflow as it reaches the disk-halo interface, rather than a result of condensation of localized cool clouds.

Truncation of thin disks at 𝑅 ≈ 𝑅 c,max
For disks fed by hot rotating inflows,  c,max corresponds to the maximum cylindrical radius at which gas cools and accretes onto the ISM (Fig. 7).This is also evident as a sharp edge of the cool gas disk at  =  c,max in the temperature maps shown in Fig. 1, and is a result of hot CGM inflows cooling and accreting only when   =  c (Fig. 4d).Similar behavior has been identified in FIRE cosmological simulations of low-redshift Milky-Way mass galaxies which are also fed by hot rotating inflows (Hafen et al. 2022;Trapp et al. 2022).
The predicted maximum accretion radius at  c,max is similar to the observed truncation radius  trunc of nearby disk galaxies, beyond which the stellar and Hi surface densities drop (e.g., van der Kruit 2007).Such a truncation is observed in 60 − 80% of thin discs at  trunc ≈ 3.5 − 4 d where  d is the disk scale length (Kregel et al.  2002; Comerón et al. 2012; Martín-Navarro et al. 2012) 6 .When a Hi warp is present it also often starts at  ≈  trunc (van der Kruit 2007).Using  trunc = 3.5 d ≈ 0.04 vir based on the  d −  vir relation from Kravtsov (2013) and  c,max ≈ 0.05 vir (eqn.15), we get  c,max ≈  trunc .It is thus plausible that observed disc truncations are ultimately a result of the abrupt cut in gas accretion beyond this radius, as suggested by Trapp et al. (2022).In the context of hot rotating CGM inflows this maximum radius of accretion is set by  c,max .

SUMMARY
In this work we derive an axisymmetric, steady-state solution for hot and rotating CGM inflows, focusing on Milky-Way mass galaxies at  ∼ 0. We demonstrate that such accretion flows transition from a spherical, hot (∼ 106 K) radial inflow to a cool (≲ 10 4 K) disk supported by rotation.This cooling occurs at the disc-halo interface, within a cylindrical radius equal to the maximum circularization radius of the flow  c,max ∼ 15 kpc and at heights || ≲ kpc above the disk.Such hot inflows are expected in the CGM if radiative cooling has dominated over feedback heating for a cooling timescale.This condition is easier to satisfy at inner CGM radii where cooling times can be substantially shorter than the Hubble time.
We find both a new analytic solution for hot inflows in the slow rotation limit, which is valid in the limit (/ c,max ) 2 ≫ 1 (section 2.4), and a numerical solution applicable also at  ≲  c,max (section 3).These solutions provide an idealized version of the hot CGM inflows identified by Hafen et al. (2022) in FIRE simulations of Milky-Way mass galaxies at  ∼ 0. The main properties of these solutions can be summarized as follows: (i) Due to a balance between radiative cooling and compressive heating, hot inflows remain hot down to where they become rotationally supported, at which point the inflow and compressive heating stop and the entire flow cools (Figs.1-4).Hot inflows thus differ qualitatively from 'precipitation' in which accretion proceeds via a subset of clouds formed due to thermal instability. 0 .Solid lines and bands correspond to medians and 16 th − 84 th percentiles.Top panels show , ,  and   , while bottom panels show  H , ⟨ / ρ⟩, / and   .The  and  panels demonstrate that cooling occurs just above the disk, at the 'disc-halo interface'.Except for the midplane flowline ( 0 = 0.5 ) the flow direction is almost vertical at the shown times, with  decreasing by 2 − 4 kpc and  changing by ≲ 0.5 kpc.At ≈ 20 Myr prior to cooling gas densities start to increase while density fluctuations become significant.This rapidly shortens the cooling time which allows the eventual drop in temperature to occur over a short timescale of ∼ 10 Myr, as can be seen in the  panel.
and Λ −22 = Λ/10 −22 erg cm 3 s −1 .Eqn. (10) treats  as the free parameter, though one can also treat the CGM mass or density profile normalization as a free parameter, and then  follows from the density relation in eqn.(10).

Figure 1 .
Figure 1.Temperature map (color) and flowlines (black lines) in a hot rotating CGM inflow.Left and middle panels show the solution in the cylindrical  −  plane and the  −  plane.The right panel depicts three specific flowlines as 3D 'tubes', where the cross-section along each tube scales as () −1 and hence illustrates the compression of the flow.Note that the hot ∼ 10 6 K phase inflows along helical paths, and cools to ∼ 10 4 K just prior to joining the ISM disk.

Figure 2 .
Figure 2. Radially-averaged kinematics and geometry of a hot rotating CGM inflow.Top: Lines show sound speed (red), rotation velocity (blue), and inflow velocity (magenta).Bottom: Black line shows the average absolute height above the midplane.Rotation velocity increases inward due to conservation of angular momentum.At radii  <  c,max where the inflow becomes fully rotation supported (  =  c ) the hot inflow cools out, the inflow halts (|  | → 0), and the geometry transitions from quasi-spherical to a disk.

Figure 3 .Figure 4 .
Figure 3. Gas properties along flowlines in hot rotating CGM inflows, versus time since a fluid element is at  = 40 kpc.Panels show spherical radius, polar angle, temperature, and rotation velocity.Different lines and bands correspond to medians and 16 th − 84 th percentiles of flowlines with different polar angles at large radii  0 .Vertical dotted lines indicate times where  drops to 10 5 K. Initially the flowlines have roughly constant  ≈  0 .About 200 Myr prior to cooling the flow geometry flattens ( → /2), and the temperature increases mainly in flowlines with small  0 .Cooling occurs when   reaches  c ≈ 200 km s −1 , i.e., at the circularization radius of the flowline  c (  0 ), indicating a transition from quasi-thermal pressure support against gravity to rotational support.Dispersion in hot gas properties prior to cooling is small, demonstrating the hot inflow is steady and axisymmetric.

Figure 5 .
Figure5.Deviations from spherical symmetry in hot rotating CGM inflows.Panels show from top to bottom the hot gas angular frequency, temperature, hydrogen density, and pressure, versus angle from the rotation axis , at  = 45 kpc (left) and  = 25 kpc (right).Black lines are based on the simulation after steady-state is achieved (the same simulation as in Figs.1-4).Magenta lines plot the analytic non-rotating solution (eqn.10), while blue lines plot the analytic slow-rotating solution (eqn.41) which accounts only for lowest-order terms in  = (/ c,max ) −2 (noted on top).In the rotating solutions density and pressure increase towards the midplane, while temperature decreases.

Figure 7 .
Figure 7. Accretion rate onto the disk versus disk radius in hot rotating inflows, for  c,max = 15 kpc and total accretion rate  = 1.7 M ⊙ yr −1 .The solid line is the accretion rate within a cylindrical disk radius  in the fiducial simulation.The step at  =  c,max corresponds to horizontal accretion from the disk edge.The dashed line is the analytic estimate, which assumes each flowline accretes at its circularization radius (eqn.48), and provides a good match to the simulation result.This predicted  disk profile in hot rotating inflows can be useful for chemical evolution models and dynamical models of galaxy disks (e.g.,Schönrich & Binney 2009;Krumholz et al. 2018).

Figure 8 .
Figure 8.The effect of turbulence on hot rotating CGM inflows.The panels show density fluctuations (top) and specific angular momentum (bottom) along flowlines, versus time since the hot CGM cools and accretes onto the ISM.Dashed lines denote the simulation with fiducial initial conditions (ICs) shown also in Figs.1-5, while solid lines denote a simulation with turbulent ICs.Line color denotes the initial polar angle of the flowline as in Fig. 4. Note the mild angular momentum loss along flowlines in the turbulent simulation, in contrast with the constant angular momentum in the fiducial simulation.
plots a projection of the line of sight velocity, weighted by the emissivity of the OVII He line, assuming CGM rotation is viewed edge on.For this calculation we use a simulation with  = 3 M ⊙ yr −1 ,  c = 200 km s −1 , and  c,max = 15 kpc.These parameters are chosen to simulate the signal around NGC 891 which has a similar circular velocity and size as the Milky-Way, but a higher SFR of ≈ 4 M ⊙ yr −1 (Popescu, C. C.

Figure 9 .
Figure9.Top: Predicted centroid shift of the OVII 0.56 keV emission line in a hot rotating CGM inflow feeding a disk galaxy.CGM rotation assumed to be viewed edge-on.Pixel size is 3 kpc × 3 kpc, corresponding to 15 ′′ resolution at a distance of 40 Mpc.The calculation is based on the fiducial simulation, with the central disk masked.Bottom: difference between lineof-sight velocity at opposite sides of the disk, in the midplane.Detecting the predicted   ∼  −1 rotation profile in the hot CGM would support the hypothesis that it is inflowing.

Figure 10 .
Figure10.X-ray emission from hot rotating inflows versus azimuthal angle (defined as the sightline orientation relative to galaxy major axis), for an edge-on galaxy.Different lines denote different impact parameters.Calculations are based on the fiducial simulation.Top: predicted soft X-ray surface brightness.Surface brightness decreases with angle due to the lower densities near the rotation axis induced by angular momentum support.Bottom: Oviii Ly/Ovii He emission line ratio.The ratio increases with angle due to higher temperatures near the rotation axis (see Fig.5).

Figure C1 .
Figure C1.Gas properties along flowlines as in Fig. 4, zoomed in on times near the time of cooling.Color denotes flowlines originating at different polar angles Total rotation completed by a fluid element in the CGM prior to accreting onto the ISM.Markers denote mean values in simulations with different  cool / ff (measured at  =  c,max , see table 1), while the thin line denotes the analytic estimate from eqn. (45).The fiducial simulation shown in Figs.1-5 is marked with a circle.Hot CGM with longer  cool have slower inflow velocities, and hence fluid elements rotate more prior to accretion.
Bregman et al. (2022)CGM by the the baryon-complete CGM mass of 0.16 halo − galaxy = 10 11 M ⊙ for  halo = 10 12 M ⊙ and a galaxy mass  galaxy = 6 • 10 10 M ⊙ .The CGM size  CGM is normalized by  vir ≈ 300 kpc.This predicted  is higher than the ≈ 1 M ⊙ yr −1 required to sustain star formation in local disks.It thus follows that for local discs to be fed by hot CGM inflows, gas originally associated with the halo must have expanded beyond  vir .Such an expanded CGM is supported by recent thermal Sunyaev-Zeldovich (tSZ) maps of nearby spirals, which indicate that the baryon budget of the halo is spread over a size of  CGM ≳ 500 kpc(Bregman et al. 2022).Using this larger estimate of  CGM in eqn.(54) would suggest  ≲ 1.5 M ⊙ , consistent with observed SFRs.The large  CGM deduced byBregman et al. (2022)thus supports the scenario that local disk galaxies are fed by hot CGM inflows.