#  Variable Modified Newtonian Mechanics from an interpolating metric II
Galactic rotational curve

C. C. Wong  Department of Electrical and Electronic Engineering, University of
Hong Kong. H.K.

###### Abstract

We use an interpolating metric from a previous work wong  with a variable
modified Newtonian potential (VMOND) to explore the issue of flatten
rotational curve of a galaxy. We examine a model of a large protogalactic
cloud with growing density perturbation, in which a central sphere collapses
while the outer protogalaxy continues to grow until density perturbation
reaches $O(1)$. This cloud then follows a Secondary Infall to collapse into a
disk which will move towards the stellar disk. The resulting mass profile
matches the mass to light ratio of the baryonic scaling model observations and
the predicted rotational curve matches the Milky Way (MW) data.

###### pacs:

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##  I Introduction

A common response to non-Newtonian rotational curve of a galaxy is that its
dynamics requires dark matter (DM) particles. Challenges remain for the
$\Lambda$CDM model in galaxy formation in which a DM halo potential has
developed before baryons arrive. During infalling the baryons receive heat
shock to virial temperature, with their angular momentum mixing with DM halo
and accrete into the galactic centre after cooling, the resulting disk size
does not match observations. This is known as the ”Angular Momentum
Catastrophe”, see vand ”. The DM mass function provides an additional tunable
potential for the flatten rotational curve but also acts as a source of
unwanted shock heating for baryon’s early gravitational collapse.
Alternatively one expects that cold streaming gas can escape heat shock from
DM halo and directly accrete onto the galactic centre and produce the correct
disk size keves . (Recent observation of coplanar dwarf galaxies around
Centaurus A muller  adds a new challenge to the $\Lambda CDM$ model.) MOND
aguirre \- mcgaugh  with its constant acceleration where Newtonian
acceleration is below a certain scale and Conformal Gravity mann0 -mann  with
a tunable parameter can provide the required non-Newtonian gravitational
potential without heat shock. Recent observations suggest something more
complicated is at work. Genzel et.al. genzel  and Lang lang  show that early
type galaxies have non-flatten rotational curves outside its scale length
$r_{0}$, which in fact fall off faster than the (Newtonian) exponential disk
from $2r_{0}\sim 5r_{0}$, where $5r_{0}$ is the disk size. By modelling DM
density profile based on the NSW model, the authors of lang  obtain the
following equation for the rotational speed $v_{rot}$, speed due to disk mass
$v_{disk}$ and speed due to dark matter $v_{DM}$

$v_{rot}^{2}(r)=v_{disk}^{2}(r)+v_{DM}^{2}(r)-2\sigma^{2}(\frac{r}{r_{0}}),$
(1)

where $\sigma$ is the velocity dispersion usually taken to be
$v_{rot}^{2}=3\sigma^{2}$ and the last term in the Eq. (1) represents pressure
support inside the disk. Using simulations, the authors in lang  reproduce the
observations by using maximal baryonic density (no DM allowed) inside the disk
and including pressure support term. One conclusion is that ” at $z\sim 2$
(around 10 Gyr look back time) there is no need for dark matter inside the
galactic disk upto $5r_{0}$ ”. Work by Simons et.al. simons  demonstrates that
from $z\sim 2$ to $z\sim 0.2$ we have
i) galactic rotational velocity growth correlates with galactic mass growth
over the said period,
ii) velocity dispersion $\sigma$ decreases sufficiently fast over cosmological
time with an universal profile for all mass range. This is also confirmed by
Kassin et. al. kassin .

Still more interestingly, for a class of galaxies called pure disk galaxies
(no bulge) sach  where observations show that central surface brightness (
proportional to the central mass density) remains unchanged over $1>z>0.4$ and
fits well with the Freeman exponential function, but over $0.4>z>0.02$ the
stellar disk mass increases upto $40\%$ while the central brightness and scale
length remain unchanged. This interesting observation suggests that early
arriving accretion of gas materials only impacts the peripheral (outer
distance $r>2r_{0}$) of the stellar disk. It is the late arriving accretions
that eventually managed to filter through the scale length boundary. Comparing
the rotational velocity and disperson of late type with early type spirals,
one notices that the rotational curves are evolving from an early Newtonian
profile to the late profile of a flat-curve over a period of around $10$ Gyr
(from $z\sim 2$.

These observations provide a serious challenge to the constant acceleration
proposals from MOND and Conformal gravity, as well as to the CDM paradigm in
which a typical CDM halo is already settled in the central region before the
baryons fall towards the centre. Continuous cold gas accretion is known to be
a crucial ingredient in the evolution of spiral galaxies, see for example
pezzulli , although accretion modelling is difficult without knowing where the
cold gas comes from. Observational support for continuous accretion is also
found for MW molla  in which $1.1\sim 2.2\times 10^{11}M_{\odot}$ accreted
material is estimated over a period of 8 Gyrs.

In a previous work wong  we find an interpolating metric between Schwarzschild
metric and Friedmann-Robertson-Walker (FRW) metrc in which a variable MOND
(called VMOND) term arises in its potential similar to Conformal Gravity,
except that this VMOND has no tunable parameter and depends strictly on the
central baryonic density and the cosmological background density. This metric
only modifies the Newtonian potential and does not introduce new scalar
particles such as from a Scalar-Tensor theory approach. We show in wong  that
VMOND can play a similar role of DM in density perturbation growth and lift
the higher CMB acoustic peaks. We show also wong3  that VMOND produces a
consistent strong lensing effect for galaxy and clusters. At late time, we
find that the VMOND is negligible comparing to a large central Newtonian
potential and does not automatically lead to flatten rotational curve.
However, it is not necessarily problematic since the early time rotational
curve is consistent with a pure central Newtonian potential. The challenge is
whether the subsequent central galactic mass growth will lead to the right
rotational curve. The origin of a constant specific angular momentum in
galaxies are thought to be due to tidal torque of galaxies from their
neighbouring protogalaixes hoyle -ryden . There are suggestions casuso  that
the origin comes from galaxy attached to the surface of large bubbles. The
general view is that once the galaxy comes free from its source, the overall
angular momentum becomes conserved. There are also observed accretion that
cold streaming gas from filaments and sheets having high specific angular
momentum burkert , but it is not clear whether these effects are critical in
galactic mass growth. Before the advent of DM, one of the most favoured
theories for galaxy formation is the Mestel theory for an initial spherical
uniform density (or constant mass) gas with constant angular momentum to
collapse into a disk. Typically, there is a central core which collapses
earlier to form an expontial disk with a constant angular speed while a
relatively small anount of late arrival gas which due to a constant mass cloud
collapses to a disk with constant rotational speed. The model is found to work
well by Crampin and Hoyle hoyle2  and Freeman freeman2 , but the challenge
remains in the high M/L (missing mass) required in the flat disk. In this
work, we propose that the mestel theory works better in conjunction with a
Secondary Infall scenario, first proposed by Gunn and Gott gunn , with a VMOND
potential. We find satisfactory match with both observed M/L and MW rotational
curve data.

In section 2, we discuss the main features of the interpolating metric and its
equation of motion in the slow speed regime. In section 3, we discuss the mass
modelling of spiral galaxies. We examine a model with a large protogalaxy with
growing density perturbation towards $O(1)$ while a central region collapses
following the Mestel theory. In section 4, we discuss the Secondary Infall of
the un-collapsed shells and the VMOND effect on such model. We compare the
resulting rotational curve prediction with MW data. The last section is a
short summary and conclusion.

##  II the model

In a previous work, we find a metric that interpolates between the
Schwarzschild metric and the Friedmann-Robertson-Walker (FRW) metric. In a
cosmological constant background this metric takes a familiar form

$ds^{2}=Zc^{2}dt^{2}-\frac{dr^{2}}{Z}-r^{2}d\phi^{2},$ (2)

where

$Z=1-\bigg{(}\sqrt{\frac{2GM}{rc2}}-\frac{H^{2}r}{c^{2}}\bigg{)}^{2},\>\>H^{2}=\frac{\Lambda}{3}.$
(3)

where $\Lambda$ is the cosmological constant.

For matter and radiation dominant epochs, the radial acceleration can be
obtained in Tolman-Lema$\hat{i}$tre metric and can be written in terms of
densities as follows

$\ddot{r}=\frac{h^{2}}{r^{3}}-\bigg{(}\frac{GM}{r^{3}}+H\sqrt{\frac{GM}{2r}}-\frac{\ddot{a}}{a}\bigg{)}r=\frac{h^{2}}{r^{3}}-\bigg{(}\rho_{b}+\rho_{b}\sqrt{\frac{\rho_{H}}{\rho_{b}}}+q\rho_{H}\bigg{)}\frac{4\pi
Gr}{3},$ (4)

where $a(t)$ is the scale factor, $\rho_{b}$ is the baryon density. We use $q$
as a book keeping parameter so that $q\rho_{H}=\rho$ is background density in
matter dominant epoch, and $q\rho_{H}=-\rho_{\Lambda}$ is background density
the in the dark energy dominant epoch. The VMOND radial acceleration is given
as

$\ddot{r}_{VM}=-Hr\sqrt{\frac{GM}{2r}}=-\frac{4\pi
Gr}{3}\sqrt{\rho_{b}\rho_{\Lambda}}.$ (5)

For a central mass system such as MW with $M=1.5\times 10^{12}M_{\odot}$, at
distance $r_{0}=10kpc$, this acceleration is $\ddot{r}=1.15\times
10^{-12}m/s^{2}$ and is $\ddot{r}=1.15\times 10^{-13}m/s^{2}$ at 1Mpc. Within
the scales of interest this VMOND acceleration is significantly less than the
Milgrom MOND acceleration of $\ddot{r}=2\times 10^{-10}m/s^{2}$. Therefore
VMOND does not directly lead to Milgrom MOND hypothesis. In Conformal Gravity
program mann0 , one obtains the metric

$ds^{2}=Z(r)dt^{2}-\frac{1}{Z(r)}dr^{2}-r^{2}d\Omega^{2},\>\>\>\>\>Z(r)=1-\bigg{(}\frac{2GM}{rc^{2}}-\gamma
r+\frac{\Lambda}{3c^{2}}r^{2}\bigg{)}.$ (6)

When comparing with the Schwarzschilds de-Sitter(SdS) metric, here $Z(r)$
takes on an additional term of the form $-\gamma r$. Deriving radial
acceleration from Eq.(6) and write in terms of densities, one has

$\ddot{r}=\frac{h^{2}}{r^{3}}-\frac{4\pi
Gr}{3}(\rho_{b}-\rho_{\Lambda})-\gamma.$ (7)

We note that the $\gamma r$ term in the conformal Gravity metric plays the
role of a constant acceleration independent of the densities due to the
central mass and cosmological background. One could tune this $\gamma$ to a
desired value in cases where Newtonian acceleration is not enough to provide
the gravitational attraction needed to fit the flatten rotational curves mann
and edery . However at this tuned value $\gamma$, the corresponding strong
lensing deflection angles for galaxies and clusters are negative which are not
compatible with observations abhatt .

By contrast, in Eq.(4) when comparing with Newtonian acceleration, the VMOND
acceleration

$\ddot{r}_{VM}=-\frac{4\pi Gr}{3}\rho_{b}\sqrt{\frac{\rho_{H}}{\rho_{b}}},$
(8)

which depends only on the ratio $\rho_{b}/\rho_{H}$ and is not tunable.

Crucial features of this model:
Comparing to the Newtonian term, VMOND is negligible when
$\rho_{b}/\rho_{H}\gg 1$ such as in a large central point mass potential.
However, effects of VMOND is completely different for $\rho_{b}/\rho_{H}\leq
1$. At the baryon overdensity region with $\rho_{b}/\rho=\delta\>(\delta\sim
10^{-5})$ at very early time, we see that the VMOND density dominates over
Newtonian gravitational attraction. For a protogalactic cloud with baryonic
density $\rho_{b}/\rho=\delta\sim O(1)$ where VMOND density plays the role of
adding a term $\rho_{b}/\sqrt{\delta}$ to the baryonic density. When an outer
gas shell of the protogalaxy comes close to the central mass after a
turnaround point, the VMOND term will diminish as $\rho_{b}/\rho_{\Lambda}$
grows larger. For $\rho_{b}/\rho_{\Lambda}\gg 1$ the VMOND density
contribution becomes negligible and Newtonian gravity dominates. Thus we can
see that the evolution of VMOND term from early time is critical to galactic
formation theory.

In wong , we show that for $\rho_{b}=\delta\rho$, in the linear perturbation
regime where the overdensity $\delta\ll 1$ the term
$\sqrt{\frac{\rho_{H}}{\rho_{b}}}=\delta^{-1/2}\gg\delta$, VMOND dominates
over Newton. We find that overdensity grows as $\delta\propto a(t)^{2}$ and
can account for the effects attributed to DM to improve the $\delta\propto
a(t)$ growth. The increasing VMOND term in radiation dominant epoch also can
account for the matter loading effects for high acoustic peaks in CMB. For
galactic rotational curve near a high mass density central potential, we have
shown above that the VMOND acceleration in Eq.(5) is much smaller than the
Newtonian acceleration whilst MOND or Conformal Gravity acceleration is larger
and constant at outside stellar disk scales, so that VMOND acceleration does
not lead automatically to a flattened rotational curve. While this VMOND
acceleration commensurates with the near stellar disk observations genzel
-lang  at early times ($z\sim 2$), the task is to explain how the rotational
curve flattens gradually over time. By incorporating the VMOND effect, we
build on the Mestel models mestel  of a spherical protogalaxy with uniform
density for which the inner region will fragment and collapses to an
exponential disk while the outer gas shells will follow a Secondary Infall
(Gunn and Gott gunn ) to reach an asymptotic mass profile similar to Mestel’s
flat disk.

##  III Galactic mass model and formation

Mass modelling for late time spiral galaxies have been studied extensively. A
review on late time disk galaxy mass modelling can be found in freeman . For
later use, here we recapulate some key features of the Freeman disk mass
modelling of Spiral galaxy up to $\sim 5r_{0}$ (hereafter we use $5r_{0}$ as
the galactic disk size) in the following: The central mass of a thin disk is
measured by its surface density which is written as

$\Sigma_{0}=\frac{M}{L}I_{0}$ (9)

where the surface density is given by $\Sigma=\int^{h_{0}}_{-h_{0}}\rho dz$,
$2h_{0}$ is the disk thickness and $I_{0}$ is the central surface brightness.
M/L is the mass to light ratio which is assumed roughly constant with radius
for different galaxy type. The surface brightness (and thus its surface
density) is found to fit by an exponential disk profile which is

$I(r)=I_{0}e^{-r/r_{0}};\>\>(\Sigma(r)=\Sigma_{0}e^{-r/r_{0}}).$ (10)

First assuming spherical symmetry and the Newtonian relation for rotational
speed $v^{2}=\frac{GM(r)}{r}$ for a centralised mass density $M(r)$, and we
have

$\frac{dM(r)}{dr}=4\pi r^{2}\rho(r)=\frac{v^{2}}{G}$ (11)

To match an observed constant density central region $\rho\sim\rho_{0}$ upto
$r=r_{0}$ we have velocity $v_{\phi}$ rises as r increases similar to a solid
body, with constant angular velocity

$\Omega=\frac{v}{r}=\sqrt{4\pi G\rho_{0}},$ (12)

which is consistent with observations.

Outside the constant density region, we expect Newtonian gravity resumes with
the mass density behaves as $\rho(r)\propto 1/r^{3}$ and
$v^{2}\propto\frac{1}{r}$.

To obtain a flat rotational curve, $v=constant$, outside the central disk at
$2r_{0}$, one needs $\rho(r)\sim 1/r^{2}\>[M(r)\sim M_{0}r]$, where $M_{0}$ is
constant. One can reproduce the constant rotational speed profile from various
potentials bovy  and the Finite Mestel disk schulz ., or from models with
protogalactic clouds such as the constant-mass Mestel sphere mestel  and the
Secondary Infall model white -nusser .

As the stellar disk exhibits a solid body behaviour, one wants to estimate its
the angular momentum. Freeman freeman2  estimates the angular momentum of the
disk, by considering stars in circular orbits at the scale length $r_{0}$ with
a centralised mass M,

$v^{2}\approx\frac{GM}{r_{0}}$ (13)

The square of angular momentum per unit mass (specific angular momentum) is
given by

$h^{2}=v^{2}\>r_{0}^{2}=\bigg{(}\frac{GM}{r_{0}}\bigg{)}r_{0}^{2}=GMr_{0}$
(14)

For a rotating exponential disk, the following relation is derived by Freeman
freeman2

$h=1.109\sqrt{GMr_{0}}$ (15)

which says that the specific angular momentum of the disk is closely connected
to the disk scale length.

The Freeman disk describes that the stellar disk is a region of constant
$\rho$ where $r<r_{0}$ $v\approx r$ which is rising as $r\rightarrow r_{0}$,
and beyond $r>2r_{0}$ one expects a newtonian fall-off $v^{2}=\frac{GM}{r}$.

Virialised galactic disk:
At late times the scale of the specific angular momentum of the virialised
Freeman disk (which is the whole galactic disk) is estimated in the form of
Eq. (15). From burkert , bullock  and mo  we use
$v_{vir}^{2}=GM_{vir}/R_{vir}$ and $v_{c}=1.2v_{vir}$ is the peak circular
velocity. For a virial radius $R_{vir}=10r_{0}\sim 13r_{0}$ and
$M_{vir}/M_{disk}=2.79$ for MW, one obtains the specific angular momentum for
the virial disk

$h^{2}=GM_{disk}l;\>\>l=1.1r_{0}\sim 1.6r_{0}.$ (16)

We see from the specific angular momentum of the virial mass of MW, which is
the Freeman disk plus HI gas disk, varies from the Freeman value
$1.109GMr_{0}$ to a slightly higher value $1.6GM_{vir}r_{0}$ at a larger
virial distance. This suggests that materials at distances further away from
the scale length have a higher specific angular momentum. This virialised
exponential Freeman disk model is largely successful upto $r\sim 2r_{0}$. The
main challenge is to explain the flatten rotation curve outside $r_{0}\sim
2r_{0}$ upto $5r_{0}$.

The simplest mass model for galactic flatten rotational curves outside $r_{0}$
can come from the ”maximal disk hypothesis” kalnajs -vanA  , which proposes
that by taking the maximal M/L ratio, one could explain the flat rotational
curves by circulating baryonic matter alone. However, although there is a
strong correlation between baryonic mass and rotational curve, works by many
(see Freeman freeman , bershady  ) etc. suggest that most spirals are
submaximal and an extra halo mass of $1\sim 2M_{b}$ is preferred palunas ,
sofue , byrun , where $M_{b}$ is baryon mass.

A largely successful ”maximal Freeman disk+HI gas disk model (baryonic scaling
model)” can be found in swaters  where they use the formula

$v_{rot}=\sqrt{\gamma v(r)^{2}_{disk}+\eta v(r)^{2}_{HI}+\epsilon
v(r)^{2}_{bulge}}.$ (17)

where $\gamma$, $\eta$ and $\epsilon$ are the mass/light ratio of the disk,
the HI gas and the bulge respectively. For bulgeless galaxies, a 2-parameter
fit is sufficient to model the rotational curves of 43 spirals and dwarfs of
both early and late type, with $\gamma$ falling within a stable maximal range
and $\eta$ in R band mostly falls in $5\sim 14$ (with an average value of 8)
range. Combined with more detailed work from the same group swaters1  the
authors find that for early spirals the HI gas mass peaks at a distance
outside $5r_{0}$, but for late spirals the mass profile of the HI gas is
gaussian and peaked around $r\sim 2r_{0}$ with a gas disk radius at
$R_{HI}\sim 5r_{0}$. The authors conclude that there is no compelling evidence
for dark particles within $2r_{0}$ and the HI gas disk size $R_{HI}$ increase
is seen to improve rotational curve flatness. A major challenge is to explain
the HI mass profiles with their peaks shown up at different distance scales at
different red-shifts. One possibility is that the HI mass peak also moves
towards the stellar disk over time. The DiskMass Survey authors also find that
using a ”Freeman disk plus HI gas plus MOND model ” on disk galaxies requires
a large M/L ratio which leads to a disk thickness that is far too large than
observed in the Disk mass Survey angus .

The early works of Hoyle hoyle , Peebles peebles , Eggen et. al. eggen
established that a galaxy such as the MW can originate from the collapse of a
large uniform density sphere protogalaxy. Mestel’s idea mestel  is to derive
equilibrium distributions of mass and angular momentum of the disk given the
conditions in the primeval sphere. In spherical coordinates $(r,\theta,\phi)$,
with $\theta$ the angle extended from the mid-plane. By considering the ratio,
based on small eccentricity and energy dissipation

$\frac{h^{2}\sin\theta/r^{3}}{GM(r)\sin\theta r^{2}}=\frac{h^{2}}{GMr},$ (18)

one notices that radial collapse will stop at some equilibrium distance $r$,
while the along the direction of the angular momentum z-axis, the collapse
will stop at a small distance. Using the poisson equation and equation of
hydrostatic support normal to the mid-plane, one obtains

$\Sigma=2\ell\tilde{\rho},\>\>\>\tilde{\rho}=\frac{\pi G\Sigma^{2}}{2c^{2}},$
(19)

where c is the sound speed, $2\ell$ is thickness of the disk, $\tilde{\rho}$
is local density independent of $z$ and the local surface density $\Sigma\sim
O(M_{tot}/\pi r_{0}^{2})$, where $r_{0}$ is the scale length and $M_{tot}$ is
the total mass. One assumes that $\tilde{z}\ll r_{0}$ as long as $GM/r_{0}\gg
c^{2}$. Then Mestel proceeds to calculate the consequences of this equilibrium
state. The equilibrium angular speed is

$\Omega^{2}=\frac{v^{2}}{r^{2}}=-\frac{1}{r}\frac{d\phi}{dr}$ (20)

where the equilibrium potential at r of a thin disk with size $R_{0}$ and
surface density $\Sigma(r)$ is given by

$\phi(r)=G\int_{0}^{2\pi}\int_{0}^{r_{0}}\frac{\Sigma(q)q}{(r^{2}+q^{2}-2rq\cos\theta)}d\theta
dq.$ (21)

For uniform density sphere, one obtains

$\Sigma(r)=\Sigma_{0}\bigg{[}1-\bigg{(}\frac{r}{r_{0}}\bigg{)}^{2}\bigg{]}^{1/2}.$
(22)

This is a limit of an uniform density spheroid of semi-axes $r_{0}$, $r_{0}$
and $(1-e^{2})^{1/2}r_{0}$, which has equilibrium constant angular velocity
$\Omega$ in the $e\rightarrow 0,\>\rho\rightarrow\infty$ limit which describes
solid body rotation where $r_{0}$ is the size of the disk with ($r<r_{0}$).
For constant mass spheroid, where density decreases slightly as radius
increases, Mestel mestel  obtains

$\Sigma(r)=\frac{\Sigma_{0}r_{0}}{r},$ (23)

which describes a flat rotational curve. Mestel also obtains a finite disk
potential, mestel  Eq.(56)

$\Sigma(r)=\frac{v^{2}}{2\pi
Gr}\bigg{(}1-\frac{2}{\pi}\sin^{-1}\frac{r}{R_{0}}\bigg{)},$ (24)

which reproduces Eq. (23) for $R_{0}\rightarrow\infty$. These Mestel’s results
received good observational supports from work of Hoyle and Crampin hoyle2
and Freeman freeman2 . Apart from an uniform isothermal sphere, there is
suggestion that one could start with a finite Mestel disk (FMD) instead and
for this FMD Schulz schulz  obtains an analytic solution which describes a
flat disk with a sharp edge. Given that the central disk is well described by
a solid body with constant angular velocity $\Omega$, we believe that an
uniform density sphere is a more appropriate starting object to collapse into
an exponential stellar disk.

The Mestel theory is recently revisited by Davies using numerical calculations
davies  in which the initial MW central disk mass is taken to be $M=3\times
10^{11}M_{\odot}$ at $z\sim 14$. The density of the cloud at $z\sim 14$ is
taken as $\rho_{0}\sim 7\times 10^{-23}kg/m^{3}$ and the critical density is
$\rho_{c}\sim 2\times 10^{-26}kg/m^{3}$ at $z\sim 0.5$. The radius of this
initial galactic region is calculated as $R_{0}=40kpc$ (we have $R_{0}=30kpc$,
if one takes $M=4\pi\rho R_{0}^{3}=2\Sigma_{0}R_{0}^{2}$ where surface density
$\Sigma_{0}=2\rho R_{0}$). The collapse time $t_{col}$ can be calculated using

$t_{col}=\sqrt{\frac{3\pi}{32G\rho}}=0.5427\frac{1}{\sqrt{G\rho}},$ (25)

which gives $t_{col}\sim 0.3\>Gyr$. For a MW virial mass taken from taylor  at
$1.2\times 10^{12}M_{\odot}$, the initial outer radus $R_{0}\sim 63kpc$. (We
note that our analysis using VMOND should require much less MW mass than the
usual virial mass quoted as there is no DM present, however using a larger
viral mass does not affect the analysis.)

##  IV Fragmentation of the centrally collapsing cloud from its outer
expanding region

The observations that star forming clouds are usually not undergoing rapid
collapse supports the view that gravitational collapse can occur in the
densest part of a cloud even when the cloud as a whole is not collapsing, see
for example larson . The Jean Mass $M_{J}$ estimate after recombination is
$M_{J}<10^{6}M_{\odot}$ so that higher density objects like globular clusters
with $M_{J}=10^{5}\sim 10^{6}M_{\odot}$ in principle can form at $z\sim 100$
when the cosmic matter density is much higher than the critical density.
Fragmentation can be further sustained by the Jean Mass relation to its
density with adiabetic index $\gamma_{0}$

$M_{J}\sim\rho^{\frac{3}{2}(\gamma_{0}-\frac{4}{3})}$ (26)

where the adiabetic index for astrophysical objects is lower than 1, so that a
localised density increase will lead to a smaller Jean Mass.

We suspect that the central region where density perturbation and radius stop
growing early while the overall cloud continues to expand, in sucg case a
large fragmentation could occur. In this case, we envisage that the overall
cloud will continue to expand following the density perturbation growth
equation until $\delta\sim O(1)$, while a central region with $\delta<1$ will
break up with the overall cloud and collapses to an exponential disk. In
tanoglidis  for a cloud of mass upto $10^{16}M_{\odot}$, analytic work and
simulation support this idea of fragmentation and they obtain a transitonal
mass scale at $10^{13}M_{\odot}$ to separate structure with two qualitatively
different cosmological evolution. An observational support for this
fragmentation scenario is in the 2015 Nature report of an observed protogalaxy
martin  in which the central disk is in place while the external cloud spans
upto a distance of $130kpc$.

###  IV.1 The Initial Condition and Evolution of the Milky Way

Next we estimate the initial condition and evolution of the Milky Way using
the idea of VMOND and a central cloud fragmentation.

Incorporating VMOND, we know that $\delta\propto a(t)^{2}$. At recombination,
we take $\delta=3\frac{\Delta T}{T}=3\times 10^{-5}a^{2}$, where
$a(z=1000)=1$, $\frac{\Delta T}{T}$ is the temperature fluctuation and $a(z)$
is the scale factor relative to the scale factor $a(z=1000)$ at recombination.
It is generally believed that galaxies can begin to form at $z\sim 100$,
$a(z)=1000/(1+z)\sim 10$, here $\delta=3\times 10^{-5}\times 10^{2}$, where
cosmic density $\rho=2\times 10^{-20}kg/m^{3}$, $\delta\rho=6\times
10^{-23}kg/m^{3}$, $\delta^{1/2}\rho=1.09\times 10^{-21}kg/m^{3}$. Under VMOND
with $\delta^{1/2}\rho$, the Jean Mass is much smaller and thus more
favourable for gravitational collapse.

To collapse to a large stellar disk the size of the MW, we need to start with
a central sphere that is significantly larger than the disk . We look at the
case, $z=9$, here $a(z)$ has grown by a factor of $100$ since recombination,
so that here we have $\delta_{0}=3\times 10^{-5}\times 10^{4}=0.3$ with
$\delta_{0}^{1/2}=0.54$, and we have an effective $\delta=0.84$ to drive
acceleration. We note that from the latest observational estimate for MW
masses $M_{r}$ (within distance $r$) from licquia  and taylor  are given as

$M_{r_{0}}=6.08\times 10^{10}M_{\odot},$ $M_{50kpc}=4.48\times
10^{11}M_{\odot}$; $M_{vir}=1.20\times 10^{12}M_{\odot}$.

To be compatible with the observed stellar disk mass, we take the inner sphere
mass $M=1\times 10^{11}M_{\odot}$ which has its initial radius $R_{I}=100kpc$
at $z\sim 9$ (Had we chosen $M=6\times 10^{10}M_{\odot}$, we have
$R_{I}=84.3\>kpc$ which may also be a good choice for our MW). For a MW mass
of $1.2\times 10^{12}M_{\odot}$, the initial outer radius is $R_{E}=228kpc$.
At this point the cosmic background density is $\rho_{0}=5.92\times
10^{-24}kg/m^{3}$. Since the effect of $\delta^{1/2}$ is to speed up the
$t_{col}$ and that $\delta^{1/2}\propto r^{-2/3}$, as $r$ decreases, based on
Newtonian gravity alone we can obtain an upper bound for collapse time for the
inner sphere by

$t_{col}=0.5427\frac{1}{\sqrt{G\rho_{0}\delta}},$ (27)

i.e. $t_{col}$ takes maximal value at $\delta=0.3$ at $t_{col}=1\>Gyr$ from
Eq, (27) by neglecting $\delta^{1/2}$. (Note: Had we chosen the collapse to
start at $z=10$, we will have a shorter upper bound collapse time $t_{col}\leq
0.47Gyr$. Without VMOND, we will have $\delta\sim 0.3\times 10^{-2}$ with a
central mass at $O(10^{8})M_{\odot}$ which may be less favourable for breaking
up with the overall cloud.) We expect that this central sphere will follow the
Mestel sphere analysis and collapses into a exponential stellar disk.

##  V Outer proto-galaxy shell density pertubation growth and turnaround

Next we follow the evolution of the outer mass shell. Here we take the simpler
approach of taking $\delta=3$ ($\frac{\Delta T}{T}\sim 1$), and follow the
protogalaxy to reach its turnaround via the energy equation. At $z\sim 9$ when
the inner spherical collapes, we have a small $\delta\sim 0.3$ and
$a(z)=10^{2}$. It is reasonable to expect the density perturbation of the
overall protogalaxy will continue to grow until it reaches $O(1)$. When $a(z)$
grows to the point $\delta=3$ we have $z=2.16$. Using the equation for radius
growth in a central mass

$r\approx\frac{1}{2}\bigg{(}GM(1+\delta^{1/2})\bigg{)}^{1/3}(6t)^{2/3}\propto
a(t).$ (28)

we obtain $a(z=2.16)/a(z=9)=3.16$. Thus up to this point the radius of the
outer shells should grow by a factor of 3.16. The outer mass shells now have a
inner shell radius at $R_{I}\sim 316kpc$ with outer shell size grown to
$R_{E}\sim 722kpc$. After this time we assume a constant mass within an
individual spherical shell, which leads to a density decrease for larger
radius shell and reaching their turnaround radius $r_{t}$ where the shell
density equals the cosmic background density. The cosmic background density is
estimated at $\rho_{0}=1.86\times 10^{-25}kg/m^{3}$ and at this point we have
$\rho_{b}=3\rho_{0}$. If the sphere collapses at this point, the upper bound
for estimated collapse time $t_{col}=2.24Gyr$.

Following the idea that this sphere can be considered as independent mass
shells, each of which will continue to expand until it is stopped by gravity.
The initial velocity $\dot{r}$ is usually taken as the cosmic background
speed,

$\frac{1}{2}\dot{r}^{2}=\frac{1}{2}H^{2}r^{2}.$ (29)

$R$ is used to denote the turnaround radius, $H_{R}$ the Hubble constant at
turnaround and the energy equation

$\frac{1}{2}\dot{r}^{2}+\frac{h^{2}}{2r^{2}}-\frac{GM}{r}-\frac{1}{2}H\sqrt{2GMr}-\frac{1}{2}H^{2}r^{2}=\frac{1}{2}\dot{R}^{2}+\frac{h^{2}}{2R^{2}}-\frac{GM}{R}-\frac{1}{2}H_{R}\sqrt{2GMR}-\frac{1}{2}H_{R}^{2}R^{2}$
(30)

For $h^{2}=1.1GMr_{0}$, at $r>R_{E}$ the angular speed term is much smaller
comparing to other terms. At $z=2.16$ we have the baryonic density
$\delta\rho(z=2.16)$ and $H=\sqrt{\frac{2GM}{\delta r^{3}}}$. The value of
$\delta$ affects significantly the time the mass shell approaches its
turnaround. If for example for large $\delta$, we can neglect the Hubble flow
and consider the mass shell in a central gravitational potential,

$\dot{r}^{2}=H^{2}r^{2}=\frac{2GM}{r}-\frac{2GM}{R}+H\sqrt{2GMr}-H_{R}\sqrt{2GMR}.$
(31)

So that $H^{2}=\frac{2GM}{3r}$ where $H_{R}$ is the Hubble constant at
turnaround and as an estimate we assume that $H_{R}^{2}=\frac{2GM}{3R}$, so
that

$R=\frac{1+1/\sqrt{3}}{1+1/\sqrt{3}-1/3}r=1.26\>r$ (32)

On the other hand if $\delta$ small, the mass shell instead will follow the
Hubble flow until the baryonic density matches the backgound density, we have
from Eq. (30)

$H_{R}^{2}R^{2}=\frac{2GM}{r}-\frac{2GM}{R}+H\sqrt{2GMr}-H_{R}\sqrt{2GMR}.$
(33)

For $H_{R}^{2}=\frac{2GM}{R^{3}}$, $H^{2}=\frac{2GM}{\delta r^{3}}$

$R=\frac{3}{1+1/\sqrt{\delta}}r=1.90\>r,\>\>for\>\delta=3.$ (34)

I) Firstly, we assume that the material that breaks away from Hubble flow, so
that the expansion stops at $\rho_{b}(R)=\frac{1}{2}\rho_{b}$ and the shell
radius will expand by another factor of 1.26 to reach their turnaround points.
Here the shell radii have grown to $R_{I}\sim 398\>kpc$ and $R_{E}\sim
909\>kpc$. The red-shift estimate at turnaround is $z\sim 0.96$ by using
Newtonian calculation only,

$\cos\eta=\sqrt{\frac{0.26}{1.26}},\>\>\>t_{c}=t_{col}\frac{2}{\pi}\bigg{(}\eta+\frac{1}{2}\sin
2\eta\bigg{)}=0.956t_{col}=3.02Gyr.$ (35)

The collapse time from the turnaround point at $z\sim 0.96$ is given by
$t_{col}=3.16\>Gyr$ and the collapse will complete at $z\sim 0.42$.

The summary of this scenario is:

At t($z=9$), the central sphere will fragment and gravitationlly collapse
fully at around $z\sim 5$ whereas the outer sphere will grow until $z=2.16$,
then it will reach its turnaround point at $z\sim 0.96$ and the sphere will
collapse onto the stellar disk at $z\sim 0.42$.

II) If the sphere follows the Hubble flow and turnaround at $R=1.9\>r$ given
above, with $R_{I}=600\>kpc$ and $R_{E}=1.37\>Mpc$, where the background
density $\rho_{H}=\rho_{b}(R)=\frac{3}{6.86}\rho(z=2.16)$, $a(z)$ has expanded
by a factor of $1.32$, so that the redshift is $z\sim 1.4$.

We use $\frac{r}{R}=\xi$, and the energy equation to calculate collapse time
$t_{col}$

$\bigg{(}\frac{dr}{dt}\bigg{)}^{2}=\frac{2GM}{r}-\frac{2GM}{R}+H\sqrt{2GMr}-H_{R}\sqrt{2GMR}+H^{2}r^{2}-H_{R}^{2}R^{2}=\frac{2GM}{R}\bigg{(}\frac{1}{c\xi}-3\bigg{)}$
(36)

$c=1/(1+\delta^{-1/2}+\delta^{-1})$

$\bigg{(}\frac{d\xi}{dt}\bigg{)}^{2}=\frac{2GM}{R^{3}}\bigg{(}\frac{1}{c\xi}-3\bigg{)}=\frac{8\pi
G\rho_{0}}{3}\bigg{(}\frac{1}{c\xi}-3\bigg{)}$ (37)

For $\delta=1$, $3c=1$

$t_{col}=\bigg{(}\frac{3}{8\pi
G\rho_{0}}\bigg{)}^{1/2}\int_{1}^{\xi}\bigg{(}\frac{\xi}{1-3c\xi}\bigg{)}^{1/2}d\xi=\bigg{(}\frac{3}{8\pi
G\rho_{0}}\bigg{)}^{1/2}\int_{1}^{\xi}\bigg{(}\frac{\xi}{1-\xi}\bigg{)}^{1/2}d\xi$
(38) $t_{col}=0.5427\sqrt{6.68}\frac{1}{\sqrt{G\rho_{b}}}=5.78\>Gyr,$ (39)

so that the end of the collapse occurs at $z\sim 0.3$.

The alternative scenario is the sphere will reach its turnaround point at
$z=1.4$ and collapse onto the stellar disk at $z=0.3$. The Infalling gas will
reach the stellar disk after dark energy dominant epoch starts at $z\sim 0.3$
and this is consistent with the pure disk galaxies observations sach .

##  VI Asymptotic behaviour of Infalling gas

The behaviour of the protogalactic shell in a secondary infall are considered
by gunn ,white \- nusser . In this model, a gas cloud of constant mass is
represented by spherical shells moving under the effect of cosmic expansion. A
mass shell in a central potential continues to move towards its turnaround
radius after which point the shell starts to follow shrinking oscillatory
cycles and moves towards its radius of closest approach. In our case, high
eccentricity shells will make only one oscillation to approach its radius of
closest approach which is smaller than the stellar disk size and thus they
will simply collapse onto the stellar disk. The gas density in the shells at
turnaround point is around 4 times the critical density. Following the work of
Nusser nusser , we consider a particle mass shell at distance $r_{i}$ and time
$t_{i}$, with enclosed mass $M_{i}$ and overdense mass $\delta M_{i}$. An
idealised choice is that these initial mass pertubations to be scale free
except at the turnaround point. One assumes that the pertubation $\delta_{i}$
take a power-law form,

$\delta_{i}=\frac{\delta
M_{i}}{M_{i}}=\bigg{(}\frac{M_{i}}{M_{l}}\bigg{)}^{-\epsilon}$ (40)

where $M_{l}$ is some reference mass scale. The equation of motion in
Newtonian potential is

$\frac{d^{2}r}{dt^{2}}=-H^{2}r-\frac{GM(r,t)}{r^{2}}+\frac{h^{2}}{r^{3}}$ (41)

where $M(r,t)=M(<r,t)$ is the mass inside radius $r$ at time $t$. where the
initial velocity of a shell is assumed to be the cosmological background
expansion rate

$\frac{dr_{i}}{dt}=Hr=\frac{2}{3t_{i}}r_{i}$ (42)

Nusser makes the assumption that

$h^{2}=2\alpha G\delta_{i}Mr_{i}=2\alpha GM_{i}r_{i}$ (43)
$e^{2}=1+4\alpha(\alpha-1)\delta_{i}^{2}.$ (44)

(note: in Newtonian gravity for a circular orbit, $\alpha=1/2$, larger
$\alpha$ value denotes higher eccentricity orbits). To obtain aymptotic
behaviour at large $t$, one further assumes that the mass distribution takes
the a power-law function of both $r$ and $t$

$M=k(t)r^{\gamma},\>\>\>\>k(t)=k_{0}t^{-s}.$ (45)

where $\gamma$ $k_{0}$, $s$ are constant and $k(t)$ is a slow varying function
of $t$. For an unit mass particle in a central potential the Lagrangian
”action” variable $J_{r}$, Nusser FG shows that for invariant $h$ and no
energy loss

$J_{r}=\int_{r_{b}}^{r_{a}}dr\bigg{(}\frac{dr}{dt}\bigg{)}=\int
dr\bigg{[}2\bigg{(}E-\frac{k(t)}{\gamma-1}r^{\gamma-1}\bigg{)}-\frac{h^{2}}{r^{3}}\bigg{]}^{1/2}$
(46) $J_{r}=2\pi
h\int_{\xi}^{1}du\bigg{[}(\xi^{-2}-1)\bigg{(}\frac{1-u^{\gamma-1}}{1-\xi^{\gamma-1}}\bigg{)}+(1-u^{-2})\bigg{]}^{1/2}$
(47)

is adiabetically invariant (slowly varying over time) where $u=1/r$. This also
implies that $\xi$ the ratio of distance of closest approach $r_{b}$ to
distance of furthest approach $r_{a}$ is constant,

$\frac{r_{a}}{r_{*}}=\bigg{(}\frac{t}{t_{*}}\bigg{)}^{q},\>\>\>\frac{r_{b}}{r_{a}}=\xi=constant,\>\>q=\frac{s}{\gamma+1}>0,$
(48)

These mass shells in principle would oscillate between $r_{a}$ and $r_{b}$
while individually they will shrink in scale over time.

$s=\frac{2}{3\epsilon}\bigg{[}\frac{\gamma-n}{n}+(\gamma-n+3)\epsilon\bigg{]},\>\>\>q=\frac{2}{3\epsilon(\gamma-n+4)}\bigg{[}\frac{\gamma-n}{n}+(\gamma-n+3)\epsilon\bigg{]}$
(49)

By defining $P(r,r_{a},t)$ to be the fraction of time a particle with further
distance $r_{a}$ spends inside $r$:

$P(v)=\frac{I(v)}{I(1)}\>\>(v\leq
1),\>\>P(v)=1\>\>(v>1);\>\>\>I(v)=\int_{0}^{v}\frac{du}{(1-u^{\gamma-n+2})^{1/2}};\bigg{(}\frac{r}{r_{t}}\bigg{)}^{\gamma-p}=\frac{1}{p}\int^{\infty}_{r/r_{t}}duu^{-(1+p)}P(u,\xi)$
(50)

In the asymptotic large t regime, one finds the exponents fillmore  when
$n=1,2,3$ denotes planar, cylindrical and spherical geometry,

$n=1$:

$\gamma=p=\frac{3}{3+\epsilon}\leq 1,\>\>\>q=\frac{4}{9};$ (51)

$n=2$:

$\gamma=1,\>\>p=\frac{3}{2+\epsilon}\geq
1,\>\>\>q=\frac{4\epsilon-1}{9\epsilon};$ (52)

$n=3$:

$\gamma=p=\frac{3}{1+3\epsilon},\>\>s=q=0,\>\>\>for\>\>\epsilon\geq\frac{2}{3}$
(53) $\gamma=1,\>\>p=\frac{6}{4+3\epsilon}\geq
1,\>\>\>s=2q,\>\>q=\frac{3\epsilon-2}{9\epsilon},\>\>\>for\>\>\epsilon<\frac{2}{3}$
(54)

We take $\gamma=1$ for the time being.

A key requirement above is the adiabetically invariant of $J_{r}$, in which
$k(t)$ is slow varying in $t$. In our case we have initially $\delta\sim 1$,
$\rho_{H}/\rho=1$ and the effective mass inside a mass shell at $R_{i}$
becomes

$M=(1+\delta^{-1/2})k(t)r^{\gamma}=\bigg{[}1+\sqrt{\frac{\rho_{H}}{\delta\rho}}\bigg{(}\frac{r}{R_{i}}\bigg{)}^{2/3}\bigg{]}k(t)r^{\gamma}=\bigg{[}1+\bigg{(}\frac{r}{R_{i}}\bigg{)}^{2/3}\bigg{]}\>k(t)r^{\gamma}.$
(55)

A typical shell would shrink from $500kpc$ to $50kpc$ while the multiplicative
factor $(1+(r/R_{i})^{2/3})$ changes from 2 to 1.215 in 5 Gyr. So that with
VMOND the effective $k(t)$ term in Eq. (46) is also slow varying function in
$t$. The arguments for the evolution of secondary infall shells above should
remain valid. We expect that the scaling behaviour and the exponents remain
the same, especially for the shells that arrive at the stellar disk in the
matter dominant epoch.

In the above analysis, the asymptotic state of the Infall mass function has
the form $M(r)\propto r^{\gamma}$. In practice, we expect that a disk will be
formed. In general it is not easy to deal with asymmetric potential bovy
analytically and one usually uses a thin disk approximation which is
numerically shown to provide a good approximation. In mestel  section 6. one
can scale-transform a nearly uniform sphere into a Mestel disk with potential
Eq.(24). Specifically it is shown that the sphere will start flattening from
the outer regions and leaving a central region going through a quasi-spherical
collapse. The asymptotic states of this mestel disk potential yields a mass
function with the same exponent $\gamma=1$. Based on this scenario, we
consider a thin disk approximation in which the mass shells moving in a thin
disk following the Nusser’s analysis, resulting in the asymptotic states of
the shells having the mass function $M(r)=M_{0}r$. Here the asymptotic shells
in radial direction does not follow an equilibrium circular path as in an
exponential disk mestel  but remains in motion moving towards its distance of
closest approach. The surface density of asymptotic disk with thickness
$2\ell$ with density $M(r)/(\pi r^{2}z)$ is thus

$\Sigma=\int_{-\ell}^{\ell}\rho dz=2\ell\frac{M_{0}r}{\pi
r^{2}}=2\ell\tilde{\rho}=\bigg{(}\frac{2\ell M_{0}}{\pi}\bigg{)}\frac{1}{r}$
(56)

which is simply the Mestel constant velocity disk at Eq. (19) and Eq. (23).
The central region of the disk will have lower density due to earlier stellar
disk collapse, and also with little radial velocity components. There is an
outer region of mass annuli with higher mass densities and radial speeds. It
is helpful to point out that from mestel  (section 6) that the constant
velocity Mestel disk solution is also not at equilibrium apart from at the
centre. The outer annuli at asymptotic state will continue to follow their
equation of motion to approach the stellar disk until some form of equilibrium
can be attained. Qualitatively this is consistent with observations in swaters
-swaters1 , where it is observed that an early time HI mass profile grows with
increasing distance and peaks outside $5r_{0}$ (we take $r_{0}=8.5kpc$ for MW
to be more precise) and the rotational curve flattens where the HI gas surface
density is flat. At late times, the mass profile inside $5r_{0}$ strongly
peaks around $2r_{0}$ and we do not have a mass distribution growing as
distance grows within $5r_{0}$. Within $5r_{0}$, this is consistent with the
consideration that there is nothing to stop the arriving mass shells to
continue to follow the equation of motion Eq. (41) and collapse onto the
stellar disk. With small central pressure, the collapse can take the form of a
shock nusser2  or mass shell accumulation bertschinger . An equilibrium can
also be achieved by the fountain mechanism proposed by Pezzulli and Fraternali
in pezzulli . We will come back to this mechanism below.

We note in wong  that in Dark energy dominant epoch, MW has a cut-off radius
of $\geq 1.3Mpc$ so that all the outer shell turnaround radii are well inside
the MW cut-off radius and no material will be lost due to the repulsive
background acceleration in the dark energy dominant epoch.

##  VII Effective angular momentum and M/L ratio

For a particle in mass shells starting at $r_{c}\sim 30r_{0}$ and coming to
$r_{0}$ scales, we have the energy equation of the metric,

$\frac{1}{2}\dot{r}^{2}+\frac{h^{2}}{2r^{2}}=\frac{(e^{2}-1)G^{2}M^{2}}{2h^{2}}+\frac{GM}{r}+\frac{1}{2}H\sqrt{2GMr}+\frac{1}{2}H^{2}r^{2}.$
(57)

For $r_{c}\gg r_{cl}$ where $r_{cl}$ is the distance of closest approach,
eccentricity e is given by $E=-GM/(r_{c}+r_{cl})\approx-GM/r_{c}$,

$e^{2}=1+\frac{2Eh^{2}}{G^{2}M^{2}}=1-\frac{2r_{0}}{r_{c}}$ (58)

As the turnaround radii asume the values $R_{E}>R_{I}\geq 398kpc$,
eccentricity $e\sim 1$, $\dot{r}=0$, and assuming negligible energy and
angular momentum losses we obtain for distance outside $r>r_{0}$

$v_{\phi}^{2}\approx\frac{2GM}{r}.$ (59)

Eq.(59) means that these collapsing mass shells close to the stellar disk have
an effectively higher rotational speed. For a single particle in a point mass
central potential, this speed would be temporary before the particle moves
back to large distances after passing its point of closest approach. However,
in this case the collapsing mass shells are still trying to reach its radius
of closest approach inside the stellar disk. The general gas density is now
given by

$\rho(r)=\frac{3}{4\pi}\frac{M(r)}{r^{3}}\propto r^{\gamma-3}.$ (60)

For high eccentricity shells to reach the inner region, if there is no low
eccentricity orbits oscillating outside $5r_{0}$, it is more appropriate to
take the cylindrical or spherical coordinates and we have $\gamma=1$.

As the radial velocity of an ellipse will slow down as a mass shell comes near
its distance of closest approach, we assume that the shock on the stellar disk
due to the mass shell arrival is weak and the eccentricity loss is minimal.
After the arrival of some mass shells to the stellar disk, the modelling of
central disk mass $M_{*}$ and stellar disk density $\rho_{*}$ with HI gas with
density $\rho_{g}$ using Eq.(60) gives

$\ddot{r}=\frac{4\pi
G}{3}\bigg{(}\rho_{*}+\rho_{g}\bigg{)}r=\frac{G}{r}\bigg{(}\frac{M_{*}}{r}+M_{0}\bigg{)}$
(61)

where $M_{0}$ only exists within the gas disk. From Eq. (61) the $M_{0}$ term
leads to a flatten rotational curve at sufficiently large r.

However, taking into account Eq.(59) Eq. (61) becomes

$v^{2}_{\phi}=\frac{2GM_{*}}{r}+2GM_{0}$ (62)

where there is also a factor of 2 for $M_{0}$ since the high eccentricity of
the collapsing outer shells also applies to the flat disk, where the gas mass
profile simply has a different form $M(r)=M_{0}r$. In this case, for the
baryonic scaling model observations swaters , the M/L requirement is reduced
by a half to $2.5\sim 7$ (or an average of 4) which is a desirable value. To
compare the analysis above with rotational curve observations,

For comparision purposes, we take a reasonable mass estimate
$M_{16kpc}=1.9\times 10^{11}M_{\odot}$ to match rotational curve data in
pbhatt  based on Eq. (57), where we have

$v_{16kpc}=321.0km/s,\>v_{obs}=321.4(\pm 25.27)km/s.$ (63)

where $v_{r}$ is calculated rotational speed at $r$ and $v_{obs}$ is
rotational speed from observation. At distances greater than $2r_{0}$
(ignoring the effect of additional mass density at larger scales) we obtain a
good match as follows,

$v_{28kpc}=245km/s,\>v_{obs}=238.0(\pm 1.54)km/s,$ (64)
$v_{50kpc}=183.6km/s,\>v_{obs}=178.5(\pm 17.6)km/s.$ (65)

The above results are depicted in Fig. 1. in which we observe that Eq. (62)
provides a good match to the rotational curves for the range $16kpc\sim 50kpc$
$(2r_{0}\sim 6r_{0}$) where gas surface density is high. Although we know that
most disks are not maximal, a question remains as to ”why assuming a maximum
M/L ratio for disk works so well?”, that is if one can raise the mass of the
central disk, a Newtonian near disk curve suffices to match observations. The
above match between Eq. (62) and observation provides an answer to this
question. (Outside $50kpc$ the surface density is much lower. The velocities
have large uncertainty at $9r_{0}\sim 10r_{0}$, but these velocities could
indicate the existence of a mass profile of the HI from Eq.(62). In Fig. 1,
similar plot including the VMOND term shows that the VMOND’s direct
gravitational effect is insignificant at these scales.)

##  VIII Accretion and radial flow

Apart from a non-Newtonian potential due to the infalling gas, at distances
close to the stellar disk, there is another dynamic processes of inflow-
outflow at play.

The basic scenario is that there is a layer $2r_{0}\sim 5r_{0}$ of high
angular speed gas circling the stellar disk and outside this layer there is
another (low metallicity) HI accretion gas layer (called hot Corona) at scales
$5r_{0}\sim 10r_{0}$ with lower angular speed with a higher temperature.

Following Pitts and Taylor pt1 -pt2  one defines the ”effective accretion
rate” surface density as

$\dot{\Sigma}_{eff}(t,r)=\frac{\partial}{\partial
t}\bigg{(}\Sigma_{*}(t,r)+\Sigma_{g}(t,r)\bigg{)}$ (66)

which is the amount of gas that is needed at a given time $t$ and a given
radius $r$ in order to sustain a given structural evolution at late times,
where $\Sigma_{*}$ and $\Sigma_{g}$ are surface density of the stellar disk
and outer H1 gas disk respectively. This fact is conveniently formalised by
the equation of conservation of mass. If, in the evolution of the gas
component, a sink term $\Sigma_{*}$, and a source term $\Sigma_{acc}$ are
accounted for, to describe star formation and accretion, respectively, then
the continuity equation can be written:

$\dot{\Sigma}_{eff}=\dot{\Sigma}_{acc}-\frac{1}{2\pi
r}\frac{\partial\mu}{\partial r}$ (67)

where

$\dot{M}=\mu=2\pi r\Sigma_{g}u_{r}$ (68)

is the radial gaseous mass flux, with $u_{r}$ being the net radial velocity of
the gas. The basic equations describing the dynamical consequences of
accretion onto a rotating disk has been introduced in the context of galaxy
evolution, by mayor  and improved by Lacey and Fall lacey  can be rewritten as

$\mu=-2\pi\alpha
r^{2}\dot{\Sigma}_{acc};\>\>u_{r}=-\frac{r}{\alpha}\frac{\dot{\Sigma}_{acc}}{\Sigma_{g}}$
(69)

where

$\alpha=\frac{h_{disk}-h_{acc}}{r\frac{\partial h_{disk}}{\partial r}}$ (70)

There is a major simplication for a flat rotational curve between the disk and
the accreting gas layer, one has

$h_{disk}=v_{disk}r;\>\>h_{acc}=v_{acc}r,\>\>\>\alpha=\frac{v_{disk}-v_{acc}}{v_{disk}}=1-\frac{v_{acc}}{v_{disk}}$
(71)

In pezzulli  in order to produce the right chemical abundance profile for MW
up to Corona scale, it is found that $\alpha$ lies in the small range
$\alpha=0.26\sim 0.3$.

The analysis in the last section provides a natural scenario of high angular
speed gas outside stellar disk at scales $r_{0}\sim 5r_{0}$. However the
rotational curve beyond $5r_{0}$ appears to support a potential of Eq. (60)
with possible angular momentum taken from high rotational velocity region.
Taking data from pbhatt  for $v_{acc}$ at $r_{acc}=8.5kpc$ and $v_{d}$ at
$r_{d}=16kpc$,

$\alpha=\frac{h_{disk}-h_{acc}}{h_{disk}}=\frac{h^{2}_{disk}-h^{2}_{acc}}{h_{disk}(h_{disk}+h_{acc})}=2\frac{h^{2}_{acc}(1-\frac{v_{acc}^{2}}{v_{d}^{2}})}{h_{acc}^{2}(1+\frac{v_{acc}}{v_{d}})}=0.318\pm
0.12.$ (72)

Although the observed velocity uncertainty is too large to confirm that
$\alpha$ does fall into the required range of $0.26-0.3$, the above
calculation demonstrates qualitatively how an out-flow can occur from $r_{0}$
toward $2r_{0}$. At a constant rotational region, one expects the outflow
radial velocity can be maintained. Based on observed rotational velocities
differences, similar mechanism can exist around $6r_{0}\sim 9r_{0}$.

##  IX Summary and Conclusion

Recent observations suggest that the early time galactic rotational curve is
consistent with Newtonian gravity and it gradually flattens as the galactic
disk grows in mass over cosmological time. Using the metric in a previous work
we explore its implication for galactic rotational curve. In a high density
centralised mass the VMOND term is negligible comparing to the Newtonian
potential so it is consistent with early time rotational curve. By following
the idea that a large protogalaxy fragments into an early collapsing Mestel
sphere with its outer cloud following a secondary infall, we find that VMOND
potential enhances the gravitational acceleration when density perturbation
$\delta$ is much less than one at high redshift and that the initial sphere
can have a much larger radius. The outer cloud which separates from the
central sphere will continue to grow in density perturbation until it reaches
$O(1)$. The Infall starts when the outer mass shells reach their turnaround
radii. In this case the outer gas density is diluted and the collapse time
increases significantly. The large time secondary infall mass profile is
similar to that of a flat Mestel disk, but the mass shells in this disk will
collapse onto the stellar disk before equilibrium is attained. Due to the high
eccentricity of the secondary infall mass shells, there is a factor of 2 added
to the mass terms in the acceleration equation leading to a higher rotational
speed. The late time mass profile of HI gas with a peak near $2r_{0}$ and the
M/L ratio improvement is consistent with the baryonic scaling model
observations. The resulting rotational curve profile matches MW data.

## References

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Fig. 1: Plot of the rotational speeds vs the distances from Milky Way centre,
where the solid line is the results from Eq.(59) for mass at $0.76\times
10^{42}$ kg, the dashed line is the Newtonian speeds and the dots with error
bars are the observational results from Ref.57 Table 2. The dotted line is a
MOND estimate at $\upsilon=220$ km/s.