\section{Introduction}

During their main sequence lifetimes, Sun-like stars host magnetised stellar winds which steadily remove angular momentum (AM) causing their rotation periods to increase with age \citep[][]{lorenzo2019constraining, nascimento2020rotation}. This process is made more efficient by the presence of a large-scale magnetic field, which transfers AM (stored in magnetic field stresses) to the stellar wind particles (proton, alphas, etc) as the winds expand \citep{weber1967angular, mestel1968magnetic}. Therefore the AM-loss rates of these stars depend strongly on the stellar magnetic field strength/geometry \citep{matt2012magnetic, garraffo2015dependence, reville2015effect, pantolmos2017magnetic, finley2017dipquad, finley2018dipquadoct}, properties which are controlled by the stellar dynamo \citep[see review of][]{brun2017magnetism} whose activity is observed to depend strongly on stellar rotation \citep{wright2011stellar, wright2016solar}. A consequence of this activity-rotation relation is the observed convergence of rotation periods for Sun-like stars during the main sequence \citep[][]{gallet2013improved, gallet2015improved}, such that for stars older than around 1Gyr there is an approximate relationship between rotation period and stellar age \citep[e.g.][]{skumanich1972time}.

This relationship between rotation and age is a valuable tool in estimating stellar ages \cite[``gyrochronology'', e.g.][]{barnes2003rotational}, and (when combined with the activity-rotation relation) provides information about the past, present, and future circumstellar environment experienced by exoplanets orbiting a given star \citep{johnstone2015evolution, gallet2019rotational, johnstone2020active}. A similar manner of rotation period evolution is broadly observed for stars with masses less than $\sim 1.3M_{\odot}$ (low-mass stars), though the time taken to converge from an initial distribution of rotation periods varies \citep{matt2015mass, garraffo2018revolution, amard2019first}. As the rotation periods of Sun-like stars (and low-mass stars) can be estimated by monitoring their brightness variations (as starspots/faculae rotate into, and out of, view), their rotation-evolution has primarily been constrained by observing open clusters with known ages \citep{agueros2011factory, agueros2017setting, mcquillan2013measuring, nunez2015linking, rebull2016rotation, covey2016rapidly, douglas2017poking, curtis2019temporary}. However, the number of open clusters with ages between 2-10Gyr is currently insufficient to constrain the rotation period-age relationship \citep[see review of][]{bouvier2014angular}. Subsequently, asteroseismology (a technique capable of determining both the star's age and rotation rate) has been used to measure the rotation periods of 21 older main sequence stars (with ages from $1-10$Gyrs), the results from which imply a weakened AM-loss rate for stars at the age of, or older than, the Sun \citep[][]{van2016weakened, metcalfe2019understanding}. In addition to these rotation period observations, an estimate of the current solar AM-loss rate can be used as an independent constraint on the rotation-age relation for Sun-like stars near the solar age \citep[see discussion in][]{finley2018effect}.

The solar wind AM-loss rate is difficult to constrain as the tangential speed of the solar wind at 1au is a few km/s \citep{nvemevcek2020non}, which is small compared to the average radial wind speed of 400-600km/s. Despite this, attempts to constrain the solar wind AM flux have been made in the past \citep[e.g.][]{lazarus1971observation, pizzo1983determination, marsch1984distribution, finley2019direct}. {Though each of these studies suffered from uncertainty either due to the precision of the spacecraft pointing or the ability of the plasma instruments to detect the small tangential flows. These uncertainties are lessened with \textit{Parker Solar Probe} (PSP) which samples the solar wind within 0.25au, where the signal to noise on the tangential wind speed measurements should be increased and the influence of wind-stream interactions reduced.} However during its first few encounters, PSP detected tangential solar wind speeds of up to $\sim 50$km/s \citep{kasper2019alfvenic}. These speeds are far greater than expected from magnetohydrodynamic (MHD) modelling \citep[e.g.][]{reville2020role} and, if they were prevalent throughout the corona, would be incompatible with the AM-loss rate implied by the slope of the rotation-age relation. \cite{finley2020solar} analysed the data from the first two orbits of PSP and showed these flows to be localised, existing along with wind streams that have large negative tangential speeds. Therefore, the average equatorial solar wind AM flux measured by PSP is similar to observations from previous spacecraft. At present, the mechanism(s) that generate these large tangential solar wind speeds have yet to be determined. One possibility is the motion/circulation of open magnetic flux at the base of the wind \citep[][]{crooker2010suprathermal, fisk2020global, macneil2020parker}, though this is also not well understood.

During PSP's first two perihelion passes (E01 and E02, respectively), neither the Solar Probe Cup \citep[SPC,][]{case2020solar} or the Solar Probe Analysers \citep[SPAN, Livi et al. submitted;][]{whittlesey2020solar} were able to confidently measure the properties of the alpha particles in the solar wind. Previously, \cite{pizzo1983determination} used \textit{Helios} observations to show that the alpha particles contain a large negative contribution to the total AM flux \citep[see also][]{marsch1984distribution}. Though, due to an error in spacecraft pointing, the data from \textit{Helios} required a significant correction, and more recent measurements of the alpha particle AM flux using the \textit{Wind} spacecraft at 1au instead suggest a much smaller (but still negative) contribution \citep{finley2019direct}. As the alphas particles carry roughly $15-40\%$ of the linear momentum in the solar wind, it is vital that their contribution to the total solar wind AM flux is better understood, in order to further constrain the slope of the rotation-age relation for Sun-like stars.

In this work, we analyse more recent SPAN-Ion observations from the third and fourth encounters of PSP (hereafter E03 and E04), which we use to determine the mechanical AM flux of both the solar wind protons and alphas particles. In addition, we use observations of the Interplanetary Magnetic Field (IMF) from the FIELDS instrument suite \citep{bale2016fields}, to evaluate the magnetic field stresses, so that the total solar wind AM flux can be determined. From the limited data available, we quantify the significance of the alpha particle AM flux in comparison to that of the protons and magnetic field stresses.

\section{Data}
\subsection{Fitting}
The SPAN-Ion instrument is an electrostatic analyser (ESA) that measures three-dimensional velocity distribution functions (VDFs) and is capable of distinguishing particle masses via time-of-flight measurements. {Three-dimensional counts spectra are organised into 32 energy-per-charge by 8 azimuthal angle by 8 polar angle bins. The 32 energy bins are logarithmically spaced from 125 eV to 20 keV, the 8 azimuthal angle bins {each} have a width of $11.25^\circ$, and the 8 polar angle bins {each} have an average width of $14.5^\circ$. It takes 0.218s to measure one complete 3D distribution. 16 proton VDFs and 32 alpha VDFs are summed together and stored in the L2 SF00 (proton) and SF01 (alpha) SPAN-Ion data products at cadences of 6.99s and 13.98s respectively.}

As of E04, a significant portion of the proton and alpha VDFs are still obscured by the spacecraft's heat shield and therefore lie outside SPAN’s field of view \citep[see Figure 8 of][]{kasper2016solar}. This leads to truncation of the VDF and significant inaccuracies in the y-components (in instrument coordinates) of the plasma moments. Since the instrument y-axis is the one most closely aligned with PSP’s tangential direction, this poses an obvious problem in calculating the AM flux. To this end, we performed bi-Maxwellian fits of the VDFs {using the Levenberg–Marquardt algorithm \citep{levenberg1944method,marquadt1963algorithm}}, in effect filling in the blocked parts of the VDF. {We parameterise a bi-Maxwellian distribution with the functional form,
\begin{equation}
f^M(\mathbf v) = n \left(\frac{m}{2\pi}\right)^{\frac{3}{2}} \sqrt{\frac{R}{T_\perp^3}} \exp \left( - \frac{\frac{1}{2}m(\mathbf{v} - \mathbf{\overline{v}})^2}{T_\perp} \left\{ \cos^2\Theta(R - 1) + 1\right\}\right),
\label{biMaxwell}
\end{equation}
where $n$ is the number density, $\mathbf{\overline{v}}$ the mean velocity, $T_\perp$ the perpendicular temperature, $R = T_\perp/T_\parallel$ the perpendicular to parallel temperature ratio, and $\Theta$ the angle between $\mathbf{v} - \mathbf{\overline{v}}$ and the magnetic field direction $\hat{\mathbf{B}}$, so that $\cos \Theta = ( \mathbf{v} - \mathbf{\overline{v}})\cdot \hat{\mathbf{B}} / |\mathbf{v} - \mathbf{\overline{v}}|$. For the protons we fit a bi-Maxwellian to both core and beam populations (denoted pc and pb respectively),
\begin{equation}
f_p(\mathbf{v}) = f^M_{pc}(\mathbf{v}) + f^M_{pb}(\mathbf{v}).
\label{protonfit}
\end{equation}}
{We constrain the proton beam component’s velocity to lie along the $\hat{\mathbf{B}}$ direction relative to the core, so that $\mathbf{v_{pb}} = \mathbf{v_{pc}} + v_d \hat{\mathbf{B}}$, where $v_d$ is the differential velocity between core and beam. There are thus 10 free parameters fit to the proton VDF. }

\subsection{Alpha Channel Contamination}
{The SF01 channel on SPAN-Ion should contain only alpha particles (as discriminated by time of flight measurements). However, a small proportion ($\sim 2-3\%$) of the protons were found to have leaked into this channel, appearing as spurious counts at lower energies, corresponding to the region of phase space occupied by the proton VDF in the SF00 channel (please see Appendix \ref{alphachannel} for some discussion of this issue). When fitting to this SF01 data product then, we include a term that corresponds to the contaminant protons, that is we fit a function of the form,
\begin{equation}
f_{\text{SF01}}(\mathbf{v}) = \epsilon f_p(\mathbf{v}) + f_{\alpha}^M(\mathbf{v}),
\end{equation}
where the first term corresponds to the spurious protons (and is the same expression as in equation (\ref{protonfit})), and the second term corresponds to a bi-Maxwellian fit (equation (\ref{biMaxwell})) to the true alpha particle component. When fitting to these SF01 counts spectra we do not fit again to the mean velocity or parallel and perpendicular temperatures of the proton core and beam components, rather we take the computed values of $\overline{v}, T_\perp$, and $T_\parallel$ from our fits to the proton beam and core, leaving them fixed, and only fit to the overall scaling (parameterized as $\epsilon$ in the expression above). It was checked that the contaminant protons did not have any energy and angle dependence, and so fitting to an overall scaling $\epsilon$ of the proton distribution was appropriate. $\epsilon$ thus represents the fraction of protons from the SF00 channel leaking into SF01 channel, which gives us a total of 7 free parameters to fit to (one for the effective contaminant density $\epsilon$, and 6 for the bi-Maxwellian describing the alpha particle distribution). An earlier method of measuring the alpha particle parameters made an attempt to first subtract the spurious protons before fitting (see Appendix \ref{alphachannel}). However we feel that the above method of fitting to both the contaminant protons and alpha particles at the same time, using fitted parameters from the SF00 channel, better accounts for overlap between the proton and alpha distributions, especially in slow solar wind. This is also important during the ubiquitous magnetic field reversals or switchbacks observed so far by PSP \citep{dudokdewit2019,mcmanus2020cross}, because the associated speed enhancements in the protons causes greater overlap of the proton and alpha VDFs in velocity space.
}

{Finally, spacecraft motion is removed from the fitted proton and alpha parameters to obtain plasma velocities in RTN coordinates. Fits were filtered for large residuals, or obviously unphysical fitted parameters, and discarded as necessary. Previous studies of the solar AM-loss have focused on the proton core population as it is thought to contain the majority of the proton AM flux, with the beam containing a small fraction of the proton mass and momentum (though this is not always the case, see Figure 3 of \citealp{stansby2020origin}). For this work we include the contribution of the beam in our calculation of the proton AM flux.}

\subsection{{Instrument Pointing Uncertainty}}
{The determination of the solar wind AM-loss rate is a very sensitive measurement, and, as pointed out by \cite{pizzo1983determination}, highly contingent on how well an instrument’s pointing direction is known. Writing a generic angular momentum particle flux term as $F_{AM} = r \rho v_r v_t = r\rho v_r^2 \tan \phi$, where $\rho$ is particle density, $r$ radial distance, $v_r,v_t$ the radial and tangential speeds respectively, and $\tan \phi = v_t/v_r$. Then the uncertainty in $F_{AM}$ due to an uncertainty in azimuthal angle $\phi$ is approximately given by $\delta F_{AM} = r\rho v_r^2\delta\phi$. An angular uncertainty $\delta \phi$ of $1^\circ$ is then seen to yield an uncertainty in $F_{AM}$ of the order of previous estimates of the solar wind AM flux. Any kind of systematic error in $\delta \phi$ will therefore pose a serious challenge to making a physically meaningful angular momentum measurement.}

{
The most restrictive constraints on instrument pointing directions on board PSP come from those required by the WISPR instrument \citep{vourlidas2016wide} to achieve its science objectives. WISPR’s pointing direction is required to be known to an accuracy of $0.1^\circ$ relative to its optical axis (John Wirzburger\footnote{PSP Spacecraft System Engineer}, Oct 2020, personal communication). In addition, the orientation of the spacecraft relative to the Sun centre is even more tightly constrained - it must be known to within $0.02^\circ$ relative to the WISPR optical z-axis. While such pointing direction constraints are not official science requirements for the SPAN-Ion instrument in the same way that they are for WISPR, the SPAN-Ion instrument team believes the pointing uncertainty to be well below $1^\circ$ (John Wirzburger, Oct 2020, personal communication). Even supposing SPAN-Ion pointing uncertainties were as much as 2-3 times larger than WISPR’s, this would still constitute an acceptable level of uncertainty allowing meaningful measurements of the solar wind AM flux to be made.
}

\subsection{Observations}
{The proton density measured by SPAN-Ion is underestimated in general compared to that measured by SPC, and needs to be scaled up to match. A required scaling up of the SPAN-Ion density is not unexpected because, of the plasma moments measured by ESA-type instruments, density is the most difficult to measure accurately since it depends most sensitively on instrumental parameters and calibration (this instrumental dependence almost entirely drops out of the expressions for higher order moments so this is not true for velocity, temperature, etc.). An instrument malfunction during E03 however meant that SPC densities were not available for most of this encounter, and E04 was the first time in the mission that the core of the proton VDF was more in SPAN-Ion’s field of view than SPC’s, thus making SPC’s density measurements in principle untrustworthy for most of the second half of E04 (as indicated by the quality flag). Because of this, and the fact that SPC densities are inter-calibrated with FIELDS quasi-thermal noise measurements, we use SPAN-Ion fitted densities of protons and alphas, scaled up by a factor of 1.25. The scaling factor should not differ between protons and alphas since the dependence on instrumental parameters is the same for both species.}
Additionally, the proton tangential speeds observed by SPAN-Ion and SPC are not always in agreement with one another, the root cause of which remains under investigation. For transparency, some comparison of the tangential wind speed in PSP's frame of reference is included within Appendix \ref{SPCvsSPAN}. As SPAN-Ion independently measures the tangential component of the flow, whereas SPC measures something closer to the flow angle in the plane of its detector plates, the velocities from SPAN-Ion are used as they are thought to be more reliable.

During E03, observations from SPAN-Ion are available from 27th August 2019 to 8th September 2019, with perihelion at $35.7R_{\odot}$ occurring on 1st September. Similarly for E04, observations cover 23rd January 2020 to 4th February 2020, with a perihelion at $27.9R_{\odot}$ (closer to the Sun by $7.8R_{\odot}$ than E03) on 29th January. When analysing quantities in this work, the data is averaged over three hour increments so that fluctuations, which are not representative of the bulk solar wind flow, and noise are reduced. Due to this averaging timescale, our choice of magnetic field data cadence from the FIELDS instrument suite has a negligible influence on our results, and so one minute cadence data is used. Any magnetic field measurements that were flagged by the instrument team as being bad/problematic are removed.

\begin{figure*}
   \begin{center}
    \includegraphics[trim=.cm .25cm .cm 0.cm, clip, width=\textwidth]{f1.pdf}
  \end{center}
   \caption{Summary of the proton core (red vertical ticks), {proton beam (green crosses),} and alpha particle (blue plus signs) properties. {The first row displays their radial velocities and densities, which influence their mass fluxes in the second row. The particle mass fluxes, when combined with their respective tangential wind speeds (shown in the third row), produce the AM fluxes in the fourth row.} In the top panels, the particle densities have the corresponding symbol shapes but are grey in colour and connected by thin coloured lines. In the bottom panels, the magnetic stresses from equation (\ref{AMequation}) are shown with {black} horizontal ticks in comparison to the particle AM fluxes. The time of perihelion during each orbit is highlighted with a green dotted vertical line. Fast wind streams (F1 and F2) and slow wind streams (S1 and S2) are highlighted in orange and cyan, respectively.}
   \label{fluxes}
\end{figure*}

\begin{figure*}
   \begin{center}
    \includegraphics[trim=.cm .25cm .cm 0.cm, clip, width=\textwidth]{f2.pdf}
  \end{center}
   \caption{The trajectory of PSP in a rotating frame of reference with the Sun, which when data are available, is coloured by the AM flux in the protons ({core plus beam,} top left), alpha particles (top right), magnetic stresses (bottom left), and their total (bottom right). Both the AM flux in the alpha particles and that stored in the magnetic stresses are multiplied by three, so that they are visible in comparison to the scale of the proton AM flux. In the background the average magnetic field polarity observed by FIELDS over one hour intervals colours parker spiral field lines, which are ballistically-mapped using the radial wind speed observed by SPC (as the data available from SPC covers a much longer time period than SPAN). A fast wind stream (F1) and a slow wind stream (S1) are highlighted in orange and cyan, respectively. The bottom right panel is zoomed-out slightly to show the extent of the observations in comparison to the system size. }
   \label{trajectoryE03}
\end{figure*}

\begin{figure*}
   \begin{center}
    \includegraphics[trim=.cm .25cm .cm .cm, clip, width=\textwidth]{f3.pdf}
  \end{center}
   \caption{Same as Figure \ref{trajectoryE03}, now with data from E04.}
   \label{trajectoryE04}
\end{figure*}

\begin{table*}
\caption{Average Properties of the Fast and Slow Wind Streams.}
\label{table:1}
\centering
\begin{tabular}{c c c | c c c | c }
\hline\hline
Wind Stream & $\langle v_{r,\alpha}-v_{r,pc}\rangle$ & $\langle r^2 F_M\rangle$ & $\langle r^2F_{AM,p}\rangle$ & $\langle r^2F_{AM,\alpha}\rangle$ & $\langle r^2F_{AM,B}\rangle$ & $\langle r^2F_{AM}\rangle$\\
 & [km/s] & [$\times 10^{11}$g/s/sterad] & [$\times 10^{30}$erg/sterad] & [$\times 10^{30}$erg/sterad] & [$\times 10^{30}$erg/sterad] & [$\times 10^{30}$erg/sterad]\\
\hline
   F1 & 98 & 1.4 & -0.38 & -0.15 & 0.10 & -0.43\\
   S1 & 19 & 1.7 & 0.06 & 0.03 & 0.07 & 0.16\\
   F2 & 82 & 1.1 & -0.44 & -0.10 & 0.09 & -0.45\\
   S2 & -3 & 2.8 & 0.24 & 0.14 & 0.07 & 0.45\\  \hline
   $\langle$Fast$\rangle$ & 90 & 1.3 & -0.40 & -0.13 & 0.09 & $-0.44$\\
   $\langle$Slow$\rangle$ & 4 & 2.5 & 0.18 & 0.11 & 0.07 & $0.36$\\
\hline
\end{tabular}
\end{table*}

\section{Solar Wind Mass Flux and Angular Momentum Flux During E03 and E04}

As the solar wind particles remove AM at a rate proportional to their mass flux multiplied by their tangential speeds, it is informative to disentangle their relative abundance from the amount of AM they transport per unit mass. Throughout most of the E03 SPAN-Ion observations, PSP was immersed in relatively fast and low density wind (until the last few days where the wind speed decreased). The radial wind speed of both the protons and alpha particles are shown in Figure \ref{fluxes}, along with their densities. The mass flux ($F_M$) of each particle species during this time are evaluated using,
\begin{equation}
    r^2 F_M = r^2 (\rho_{pc} v_{r,pc} + \rho_{pb} v_{r,pb}) + r^2 \rho_{\alpha} v_{r,\alpha},
    \label{massflux}
\end{equation}
which is the sum of mass flux in the proton, and alpha particles multiplied by $r^2$ (corresponding to the flow of mass per solid angle). Here $r$ is the radial distance of PSP from the Sun, and the subscripts $pc$, $pb$, and $\alpha$ distinguish the proton core, proton beam, and alpha particle properties of density $\rho$, and radial wind speed $v_r$. The mass flux in each term of equation (\ref{massflux}) is displayed in Figure \ref{fluxes}. During E03, the proton mass flux remained roughly constant at $\sim 1.1\times 10^{11}$g/s/steradian {(with the beam carrying on average $\sim 31\%$ of the proton mass flux)}, whilst the alpha particle mass flux varied from $0.08-0.4\times 10^{11}$g/s/steradian. In the E04 SPAN-Ion observations, PSP encountered more varied solar wind conditions than E03 (this is also reflected in the IMF polarity during E04). Initially PSP was immersed in fast wind similar to that of E03, but then transitioned into slower and denser wind as PSP neared perihelion. Accordingly, the proton mass flux, as evaluated from equation (\ref{massflux}), increased two-fold. The alpha particle mass flux during E04 has some interesting features, including a large decrease prior to perihelion.

Combining the observations from SPAN-Ion and FIELDS, the solar wind AM flux ($F_{AM}$) is evaluated as a sum of the mechanical AM carried by the protons ($F_{AM,p}$), and alpha particles ($F_{AM,\alpha}$), along with a term prescribing the transfer of AM through magnetic field stresses ($F_{AM,B}$). This is given by,
\begin{align}
r^2F_{AM}=& r^2F_{AM,p} + r^2F_{AM,\alpha} + r^2F_{AM,B},\nonumber\\
=& r^3\sin\theta(\rho_{pc} v_{r,pc} v_{t,pc} + \rho_{pb} v_{r,pb} v_{t,pb}) + r^3\sin\theta\rho_{\alpha} v_{r,\alpha} v_{t,\alpha} \nonumber \\ &-r^3\sin\theta\frac{B_t B_r}{4\pi},
\label{AMequation}
\end{align}
where $\theta$ is its colatitude of PSP, $v_t$ denotes the tangential wind speed of each particle, $B_r$ is the radial magnetic field strength, and $B_t$ is the tangential magnetic field strength. A factor of $r^2$ is included, similarly to the mass flux in equation (\ref{massflux}), in order to remove any dependence of the AM flux on radial distance (note this quantity is now the flow of AM per solid angle, however it continues to be referred to as an AM flux for simplicity). {The tangential wind speeds of each particle species are shown in Figure \ref{fluxes}.} The resulting solar wind AM flux carried by each particle species during both E03 and E04 are shown in the bottom panels of Figure \ref{fluxes}, and in the context of PSP's location (in the rotating frame of the Sun) in Figures \ref{trajectoryE03} and \ref{trajectoryE04}. Note that PSP's latitude varies by a few degrees leading up to perihelion (heliographic latitude of around $-4^{\circ}$), which is not shown in Figures \ref{trajectoryE03} and \ref{trajectoryE04}. The structure of the IMF is depicted in the background by parker spiral magnetic field lines spaced an hour apart along PSP's trajectory which are ballistically-mapped from PSP using the average radial wind speed observed by SPC, and coloured by the average polarity of $B_r$ from FIELDS. The lower right panel of Figure \ref{trajectoryE03} indicates that during E03, PSP was broadly connected to the same side (negative $B_r$) of the Heliospheric Current Sheet (HCS). Whereas in Figure \ref{trajectoryE04}, the polarity of the IMF is much more variable, suggesting multiple crossings of the HCS. This is reflected in the range of wind speeds PSP encountered during E04, and the fluctuating abundances of the protons and alpha particles.

As previously noted in \cite{finley2020solar}, the magnetic stress term shows little variation compared to the proton AM flux. Figure \ref{fluxes} shows that, in general, the proton core carries the largest positive and negative AM fluxes, with the {proton beam and} alpha particle contributions being much less (in large part due to their lower abundance), and typically having the same sign as the proton core. {The proton beam has a similar behaviour in velocity to the alpha particles (and a similar magnitude of mass flux), therefore its contribution to the total AM flux is similar in size to that of the alpha particles. However, it is worth noting that the proton beam is the least well constrained component in the fitting process, and so throughout this work the total proton AM flux is favoured for comparison. The ratio of the alpha particle AM flux to the total mechanical AM flux, i.e. both the protons (core plus beam) and the alpha particles, is displayed as a histogram in Figure \ref{ratio}. Based on the observations from both E03 and E04, the alpha particles carry around $20\%$ of the mechanical AM in the solar wind (mean values of $19\%$ in E03, and $23\%$ in E04). Their contribution can vary significantly up to $50 - 80\%$, though there is a large number of observations with the alpha particles carrying less than $10\%$ of the total mechanical AM flux. }

\begin{figure}
   \begin{center}
    \includegraphics[trim=.cm .25cm .cm .25cm, clip, width=0.45\textwidth]{f4.pdf}
  \end{center}
   \caption{Frequency of observing a given ratio of the alpha particle AM flux to the total mechanical AM flux {(proton core, proton beam, and alpha particles)} in the solar wind during E03 (top) and E04 (bottom). The average values from E03 and E04 are annotated and shown with red dashed lines. The alpha particles carry around a fifth of the mechanical AM flux in the solar wind. }
   \label{ratio}
\end{figure}

\begin{figure*}
   \begin{center}
    \includegraphics[trim=.35cm .25cm .35cm .cm, clip, width=0.9\textwidth]{f5.pdf}
  \end{center}
   \caption{Distribution of radial and tangential speeds for the solar wind protons and alpha particles. All the E03 and E04 observations of the proton core (vertical ticks), and alpha particles (plus signs) from SPAN-Ion are plotted in the left panel in grey, with the fast (F1 and F2) and slow (S1 and S2) wind streams highlighted. The distributions from the fast and slow wind streams are shown separately in the middle and right panels, respectively. {These include the proton beam (crosses) for completeness.} In each case, the average difference vector between the alpha particle and proton {core} velocities in the RT-plane are annotated {(shifted away from the data)}.}
   \label{velocityComp}
\end{figure*}

\section{Behaviour of the Alpha Particle Angular Momentum Flux}

\subsection{Fast Wind versus Slow Wind}
During the combined observations of E03 and E04, PSP was immersed in relatively fast wind ($v_r>450$km/s) more often than slow wind. During each encounter, one fast and one slow wind stream are identified (in orange and cyan, respectively) in order to further examine trends in the alpha particle AM flux. These time periods are highlighted, and numbered, in Figures \ref{fluxes}, \ref{trajectoryE03}, \ref{trajectoryE04} and \ref{ratio}. The averaged quantities from each stream are detailed in Table \ref{table:1}. {Figure \ref{ratio} shows that there is no consistent trend in the ratio of AM carried by the alpha particles (to the total mechanical AM flux) between the fast and slow streams.} On average in the two fast wind streams identified (F1 and F2), both the protons and the alpha particles are observed to carry a negative AM flux (i.e. adding AM to the Sun). {The distribution of the radial and tangential wind speeds of the protons and alpha particles are shown in detail in Figure \ref{velocityComp}.} The average tangential speeds in the fast wind are $\langle v_{t,pc}\rangle=-12$km/s, {$\langle v_{t,pb}\rangle=-26$km/s,} and $\langle v_{t,\alpha}\rangle=-26$km/s. In contrast to this, the slow wind streams (S1 and S2) appear to have a mostly positive AM flux in the protons and alpha particles (i.e. removing AM from the Sun, as expected). In this case their average tangential speeds are $\langle v_{t,pc}\rangle=5$km/s, {$\langle v_{t,pb}\rangle=-4$km/s,} and $\langle v_{t,\alpha}\rangle=12$km/s. In either wind conditions, the magnetic stresses are roughly the same and are a positive contribution to the total.

It is unlikely that the fast wind carries a net negative AM flux everywhere in the heliosphere, instead our observations are likely influenced by the large spatial and/or temporal variations in the equatorial AM flux \citep{finley2020solar}. Though a recent study of the solar wind at 1au by \cite{nvemevcek2020non} was able to show a similar correlation with the tangential speeds of the fast and slow wind \citep[see also Figure 2 of][]{finley2019direct}. Therefore, we cannot rule out the possibility that this dichotomy in AM fluxes (between fast and slow wind streams) exists more generally in the equatorial solar wind. Once more data has been collected by PSP, a statistical study of the AM flux carried by the fast and slow wind could determine the validity of this hypothesis. If this dichotomy were to exist, matching the large AM flux differences between the fast and slow wind would be a new observational constraint on models that aim to reproduce the solar wind acceleration and propagation \citep[e.g.][]{reville2020role, shoda2020alfv}. Currently, the source(s) of these enhanced tangential wind speeds remain unclear. As PSP moves closer to the Sun it becomes less likely that these features are a result of wind-stream interactions, and so perhaps a mechanism nearer the base of the wind is responsible, such as the motion of open magnetic flux in the photosphere \citep[e.g.][]{fisk2020global}.

\subsection{Alpha-Proton Differential Velocity}
The differential velocity between the alpha particles and the proton {core} (${\bf v_{\alpha} - v_{pc}}$), is observed to vary significantly between the fast and slow wind streams studied in this work. In the fast wind streams, the radial speed difference between the two particle species is around 90km/s, whereas for the slow wind streams its is nearer 5km/s. Previous observations of the solar wind at various distances from the Sun have shown that ${\bf v_{\alpha} - v_{pc}}$ is a function of wind speed, such that at low speeds the protons and alpha particles have similar velocities \citep{marsch1982solar, neugebauer1996ulysses}. At high speeds, ${\bf v_{\alpha} - v_{pc}}$ grows larger and becomes increasingly aligned with the direction of the IMF \citep{asbridge1976helium}. {Figure \ref{velocityComp} shows the sampled velocity space of the alpha particles and protons. Indeed the velocities of the alpha particles and proton {core} in the slow wind are far more similar than that of the fast wind. The average difference vector for the fast and slow streams are over-plotted in black.}

{Figure \ref{DiffVelocity} shows the direction of ${\bf v_{\alpha} - v_{pc}}$ in the RT-plane for each three hour average, along with the IMF direction (both of which are around $10^{\circ}$ from radial). The right panel of Figure \ref{DiffVelocity} shows the differential velocity vectors from the left panel with respect to the IMF direction from the middle panel. Given the relationship between the differential velocity and the IMF direction, it may be possible to infer the contribution of the alpha particles to the solar wind AM flux by rewriting $F_{AM,\alpha}$ as,}
\begin{equation}
    r^2 F_{AM,\alpha}= r^3\sin\theta \bigg(\cfrac{\rho_{\alpha}}{\rho_{pc}}\bigg)\rho_{pc}(v_{r,pc}+\delta v_{\alpha p, r})(v_{t,pc}+\delta v_{\alpha p, t}),
\end{equation}
{where $\delta v_{\alpha p, r}$ and $\delta v_{\alpha p, t}$ are the radial and tangential components of the differential velocity vector ${\bf v_{\alpha} - v_{pc}}$. Such that, given the abundance $\rho_{\alpha}/\rho_{pc}$ and a relation for ${\bf v_{\alpha} - v_{pc}}$ based on more readily observed parameters, e.g. ${\bf v_{\alpha} - v_{pc}} \approx f({\bf B},{\bf v_{pc}},...)$, the alpha particle AM flux could be estimated during times when they are not directly observed. However, as there are only a small number of observations available at present, it is difficult to say how reliable such a relationship for ${\bf v_{\alpha} - v_{pc}}$ would be. For F1 and F2, the angle between ${\bf v_{\alpha} - v_{pc}}$ is more strongly aligned with the IMF direction than during S1 and S2 (this can easily be seen in the right panel of Figure \ref{DiffVelocity}). Though the larger range of angles observed between ${\bf v_{\alpha} - v_{pc}}$ and the IMF during the slow wind streams is likely a consequence of small fluctuations between the particle speeds (including noise), as their velocity vectors are similar in size {due to the increased collisionality of the slow wind in comparison to the fast wind \citep[see][]{alterman2018comparison}}. Note that the proton beam generally has a similar velocity distribution in the R-T plane to that of the alpha particles\footnote{This is in-part because the differential velocity between the proton core and beam was previously constrained to lie along the magnetic field direction (see Section 2.1).}, and so when observations of the proton beam are available they too could be used to reconstruct the alpha particle AM flux. Though fits to the proton beam often carry higher uncertainties than for the proton core (due to the smaller number of counts in that region of phase space), and the proton beam population may not be present if the proton distribution is close to being isotropic.}

\begin{figure*}
   \begin{center}
    \includegraphics[trim=.3cm .25cm .5cm .cm, clip, width=\textwidth]{f6.pdf}
  \end{center}
   \caption{Left: Distribution of the differential velocity vector between the alpha particles and proton {core} in the RT-plane. Middle: Distribution of IMF vectors in the RT-plane. Right: Distribution of the differential velocity vector between the alpha particles and proton {core} with respect to the IMF. The IMF points along the y-axis, the sign of the IMF (inward or outward from the Sun) is inconsequential. All observations are averaged over three hour intervals, as done previously when calculating the AM flux. Data from the fast (F1 and F2) and slow (S1 and S2) wind streams are highlighted in orange and cyan, respectively. }
   \label{DiffVelocity}
\end{figure*}

Consider the consequence of the differential velocity aligning with the IMF for a fast wind stream with a positive AM flux in the proton {core} (which is not observed here, but likely exists). For the differential velocity to be aligned with the IMF the alpha particles would have to take a smaller or negative tangential velocity. Though this has yet to be observed by PSP, it is capable of explaining previous observations from \textit{Helios} in which a positive AM flux in the proton core was observed with a negative AM flux in the alpha particles \citep{pizzo1983determination}.  Moreover, as the differential velocity is observed to lessen as the solar wind travels through interplanetary space \citep[see][]{neugebauer1996ulysses}, it is likely that AM is exchanged between the two species. So this could explain the observations from \textit{Wind} at 1au which show the alpha particles to have a much smaller contribution to the total AM flux \citep{finley2019direct}. However, the mechanism which produces the differential velocity between the alpha particles and the proton core is still unknown.

\section{Conclusion}

Using data collected by the SPAN-Ion and FIELDS instruments on-board PSP during its third and fourth closest approaches to the Sun, the strength of the AM flux carried by the alpha particles has been evaluated. Typically the alpha particles host around a fifth of the mechanical AM flux contained within the solar wind particles, the rest is carried by proton {core and beam}. Though it is difficult to say if this result is representative of the globally-averaged solar wind. The sign of the alpha particle flux follows closely that of the proton {core}. The fast wind is observed to carry a net negative AM flux, and the slow wind is found with a comparable but positive AM flux. However given the small number of available observations, it is difficult to say if this dichotomy is typical of the equatorial solar wind, or simply a reflection of the variability of the solar wind AM flux \citep[as discussed in][]{finley2020solar}.

For the data presented, there is a strong preference for the differential velocity between the alpha particles and proton {core} to be aligned with the IMF in the fast solar wind. For the slow wind, the alpha particles and protons are observed have more similar velocities. This was previously observed with \textit{IMP 6 $\&$ 7} \citep{asbridge1976helium}, \textit{Helios} \citep{marsch1982solar}, and \textit{Ulysses} \citep{neugebauer1996ulysses}. The evolution of the differential velocity with radial distance likely explains some of the differences between previous observations from \textit{Helios} \citep{pizzo1983determination}, and \textit{Wind} \citep{finley2019direct}. In future, trends in this quantity could be used to constrain estimates of the AM flux in the alpha particles when only the proton {core} is detected. Additionally, the proton beam velocity is shown to be similar to that of the alpha particles, so this too can be used to constrain the alpha particle AM flux (though measurements of the proton beam are typically more uncertain than the proton core). Interestingly, this work suggests that the proton beam contains a similar magnitude of AM flux to the alpha particles, and so (like the alpha particles) the beam should also be included in future estimations of the solar wind AM flux.

The average AM flux computed in this work would suggest that the Sun is gaining AM (i.e. spinning up). However this seems unrealistic, and is more likely due to: 1) PSP sampling only a small fraction of the equatorial wind, and 2) PSP primarily sampling the fast solar wind which is more often subject to dynamical interactions (i.e. deflections or compression/rarefactions {that generate a tangential flow in the solar wind) than the slow wind \citep[this can be seen in 1au measurements; e.g.,][]{finley2019direct, nvemevcek2020non}.}
Using a larger set of observations from PSP during its first two orbits (E01 and E02), the solar wind AM-loss rate (in just the protons and magnetic field stresses) was previously estimated to be $2.6\times10^{30}$erg and $4.2\times10^{30}$erg, respectively \citep{finley2020solar}. Assuming the observed fraction of the mechanical AM flux in the alpha particles is representative of the solar wind in general, then this previous estimate can be re-evaluated to include the alpha particle contribution {by increasing the proton AM flux by a quarter. As the ratio of the proton AM flux to magnetic stresses varied between orbits, this leads to an increase in the solar AM-loss rate of $12\%$ for E01, and $19\%$ for E02 (values of $2.9\times10^{30}$erg and $5.0\times10^{30}$erg, respectively).} So despite including the alpha particles, our estimate of the solar wind AM-loss rate remains smaller than the value of $6.2\times10^{30}$erg expected from a \cite{skumanich1972time}-like rotation-age relation for Sun-like stars \citep[see][]{finley2018effect}. Thus the slope of the rotation-age relation at the age of the Sun is currently steeper in rotation-evolution models than would be implied from observations of the solar wind.

With the solar cycle progressing towards increased solar activity, future works will be able to observe how the contribution of the alpha particles changes as their abundance increases \citep{alterman2020helium}. In addition, more frequent observation of the solar wind by both PSP, and \textit{Solar Orbiter} \citep[][]{mueller2013solar, muller2020solar, zouganelis2020solar}, will bring us closer to understanding the mechanisms that distribute the solar wind AM flux across the protons, alpha particles, and magnetic field stresses. \textit{Solar Orbiter} will also be able to sample the AM flux away from the solar equator, which will also help to constrain the solar AM-loss rate (as our current estimates rely on measurements of just the equatorial AM flux).

\begin{acknowledgements}
    {We thank the anonymous referee for their constructive comments which resulted in a more thorough discussion of the data products, the resulting velocity distributions, and the angular momentum flux carried by the proton beam population. These changes greatly increased the overall quality of this paper.}
    In addition, we thank the SWEAP and FIELDS instrument teams of Parker Solar Probe, and the NASA/GSFC's Space Physics Data Facility for providing this data.
    Parker Solar Probe was designed, built, and is now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s Living with a Star (LWS) program (contract NNN06AA01C). Support from the LWS management and technical team has played a critical role in the success of the Parker Solar Probe mission.
    AJF and SPM acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682393 AWESoMeStars).
    Figures in this work are produced using the python package matplotlib \citep{hunter2007matplotlib}.
\end{acknowledgements}

\bibliographystyle{aa}


\section{Introduction}
The ground state of a
 condensed cloud of fermionic matter,
 interacting only gravitationally
 and having a
 mass $M$ below the Oppenheimer-Volkoff (OV) limit \cite{opp},
 is a cold fermion star
 in which the degeneracy pressure
balances the gravitational attraction of the fermions.
Degenerate stars of fermions in the mass range between
  10  and  25 keV are particularly
interesting~\cite{vio}, as they could explain, without resorting
to the black-hole hypothesis, at least some of the features
 observed around the supermassive compact dark objects
 with masses in the range of
 $M=10^{6.5}$ to $10^{9.5}$ solar masses, that are
reported to exist at the centers of a number of
galaxies~\cite{Tonr,Dres10,Dres11,Korm12,Korm13,Korm14}, including
our own~\cite{Lacy15,Lacy16}, and quasistellar objects (QSO)
\cite{Bell,Zeld,Blan,Bege}.  Indeed, there is little difference
between a supermassive black hole and a fermion star of the same
mass near the OV limit, a few Schwarzschild radii away
from the object \cite{bil2,tsi}.

The purpose of
this paper is to study,
in the framework
of a general-relativistic Thomas-Fermi model,
 the formation of such a star
 that could have taken place
 in the early universe
 shortly after the nonrelativistic
 fermionic matter began to dominate the radiation.
 This system was previously studied in the
 Newtonian approximation~\cite{thi,her1,her2,bau,mes,bil1},
 and it was shown that the
  canonical and grand-canonical ensembles
     for such a system
have a nontrivial thermodynamical limit~\cite{thi,her1,her2}.
Under certain conditions these systems
 will undergo a phase transition
 that is accompanied by a gravitational collapse~\cite{mes,bil1}.
 The phase transition
occurs uniquely
in the case of
 the attractive gravitational interaction of
neutral fermions.
As the phase transition does not happen for particles
obeying Bose-Einstein or Boltzmann statistics,
this phenomenon is quite distinct from the usual gravitational
clustering of collisionless dark-matter particles.
Gravitational
condensation will also take place if the  fermions have an
additional short-range weak interaction, as neutrinos, neutralinos,
gravitinos,
 and other
weakly interacting massive particles do.

 Effects of general relativity
 cannot be neglected when
 the total mass of the system is close to the
 OV limit~\cite{opp}.
 There are three main features that distinguish the
 general-relativistic
 Thomas-Fermi model
  from the Newtonian one:
  {\it i}) the equation of state is relativistic,
  {\it ii}) the temperature and chemical potential are
  metric-dependent local quantities, and
  {\it iii}) the gravitational potential satisfies Einstein's field
  equations instead of Poisson's equation.

This paper is organized as follows:
In Section~\ref{sec2} we briefly discuss
the nonrelativistic Thomas-Fermi model
at finite temperature.
In Section~\ref{sec3}  this model is extended within
a general-relativistic framework.
 In Section~\ref{sec4} we discuss the
solution at zero and finite
 temperature  and, in particular,
the conditions under which
the first-order gravitational phase transition occurs.
Conclusions are drawn
in Section~\ref{sec6}
and, finally, in Appendix~\ref{app1}, we prove a theorem on
the extremal properties of the free energy.

\section{Thomas-Fermi model in Newtonian gravity}
\label{sec2}

Consider a system of $N$ fermions
of mass $m$
interacting only gravitationally, confined in
a spherical cavity of radius $R$, in equilibrium
at a finite temperature $T$.
For large $N$, we can
assume that the fermions move in a spherically
 symmetric
 mean-field potential
$V(r)$ which satisfies Poisson's equation
\begin{equation}\label{eq00}
\frac{1}{r}\frac{d^2}{dr^2}(rV) = 4\pi G  m^2 n,
\end{equation}
$G$ being the gravitational constant.
The number density  of the
fermions (including antifermions) $n$ can be expressed
in terms of the Fermi-Dirac distribution
(in units $\hbar=c=k=1$)
\begin{equation}\label{eq02}
n(r) = g \int \frac{d^3q}{(2\pi)^3}
\left[1+\exp \left(\frac{q^2}{2mT}+\frac{V(r)}{T}-\frac{\mu}{T}
\right)\right]^{-1}.
\end{equation}
Here
$g$ denotes the combined spin-degeneracy factors of
the neutral
fermions and antifermions, i.e., $g$ is 2 or 4 for Majorana
 or Dirac fermions, respectively.
 For each solution
 $V(r)$ of Eq.~(\ref{eq00}),
the chemical potential $\mu$ is adjusted so that
 the constraint
\begin{equation}\label{eq04}
\int d^3r n(r) = N
\end{equation}
is satisfied.
It may be shown that a particular spherically symmetric
configuration $\bar{n}(r)$ will satisfy
Eqs.~(\ref{eq00})-(\ref{eq04})
if and only if it extremizes the free
 energy functional
  defined as~\cite{her1}
\begin{eqnarray}
F[n]
\! & \! = \! & \!
     \mu[n] N     -
 \frac{1}{2}
\int d^3r n(r) V[n]
\nonumber  \\
     &  &
- Tg  \int\frac{d^3rd^3q}{(2\pi)^3}
\ln\left( 1+\exp \left(-\frac{q^2}{2mT}-
\frac{V[n]}{T}+\frac{\mu[n]}{T}\right)\right),
\label{eq06}
\end{eqnarray}
where $V[n]$ and $\mu[n]$ are
implicit functionals of $n(r)$ through
Eqs.~(\ref{eq00})-(\ref{eq04}).
For a physical
solution, we have to require that the free energy is
minimal, i.e.,
\begin{equation}\label{eq08}
\left. \frac{\delta F}{\delta n}\right|_{\bar{n}} = 0,
\;\;\;\;
\left. \frac{\delta^2 F}{\delta n^2}\right|_{\bar{n}} \geq 0 .
\end{equation}
The set of self-consistency equations
 (\ref{eq00})-(\ref{eq04}), together with
 (\ref{eq08}), comprises the
 nonrelativistic
 gravitational Thomas-Fermi model.

It may be easily shown that
the following scaling property holds:
  If the potential energy
 $V(r)$ is a solution to the self-consistency
 equations (\ref{eq00})-(\ref{eq04}),
  then the rescaled
 $\tilde{V}=A^4 V(Ar)$,
 with $A>0$,
 is also a solution with
 the rescaled
 fermion number
 $\tilde{N}=A^{3}N$,
 radius
 $\tilde{R}=A^{-1}R$,
 and temperature
 $\tilde{T}=A^4T$.
 This property, which will be  referred to as
 {\em nonrelativistic scaling},
 implies the existence of
 a thermodynamic limit
 of
 $N^{-7/3}F$,
 with
 $N^{1/3}R$ and
 $N^{-4/3}T$
 approaching constant values
  for
 $N\rightarrow\infty$.
 In this limit,
 the Thomas-Fermi equation becomes exact
 \cite{her1,her2}.

\section{Thomas-Fermi model in general relativity}
\label{sec3}

Consider a self-gravitating
gas consisting of $N$ fermions of mass
$m$ in equilibrium within a sphere of radius $R$.
We denote by
$p$, $\rho$, $n$, and $\sigma$ the  pressure,
energy density, particle number density, and
entropy density of the gas, respectively.
The metric generated by the mass distribution
is  static, spherically symmetric, and asymptotically
flat, i.e.,
\begin{equation}
ds^2=\xi^2 dt^2 -(1-2{\cal{M}}/r)^{-1} dr^2 -
     r^2(d\theta^2+\sin \theta d\phi^2).
\label{eq13}
\end{equation}
 Einstein's field equations are then given by
\begin{equation}
\frac{d\xi}{dr}=\xi\frac{{\cal M}+4\pi r^3 p}{r(r-2{\cal{M}})} \, ,
\label{eq40}
\end{equation}
\begin{equation}
\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho,
\label{eq42}
\end{equation}
with the boundary conditions
\begin{equation}
\xi(R)=\left(1-\frac{2{\cal{M}}(R)}{R}\right)^{1/2}
\, ; \;\;\;\;\;
{\cal{M}}(0)=0.
\label{eq44}
\end{equation}
The  equation of state may be represented
in a parametric form \cite{ehl}
\begin{equation}
n = g \int \frac{d^3q}{(2\pi)^3}\,
\frac{1}{1+e^{E/\bar{T}-\bar{\mu}/\bar{T}}} \, ,
\label{eq30}
\end{equation}
\begin{equation}
\rho = g \int \frac{d^3q}{(2\pi)^3}\,
\frac{E}{1+e^{E/\bar{T}-\bar{\mu}/\bar{T}}} \, ,
\label{eq31}
\end{equation}
\begin{equation}
p = g \bar{T} \int \frac{d^3q}{(2\pi)^3}\,
\ln (1+e^{-E/\bar{T}+\bar{\mu}/\bar{T}}) \, ,
\label{eq32}
\end{equation}
where
$g$ denotes the spin degeneracy factor
 and $E=\sqrt{m^2+q^2}$.
 The quantities
$\bar{T}$ and $\bar{\mu}$ are
the local temperature and chemical potential,
respectively.
As discussed in Appendix \ref{app1},
thermodynamic and hydrostatic equilibrium
in the presence of gravity
implies
\begin{equation}
\bar{T}(r) =\frac{T}{\xi(r)}\, ; \;\;\;\;\;\;
\bar{\mu}(r)=\frac{\mu}{\xi(r)} \, .
\label{eq24a}
\end{equation}
The constants
$T$ and $\mu$  are the
temperature and chemical potential at infinity.
Although the matter is absent at $r=\infty$,
the temperature at infinity has
  a physical
meaning: T is the "red shifted" temperature
\cite{bar}
 of the black-body radiation of a gravitating object
 in equilibrium at finite temperature
measured at infinity.
In other words, our gravitating object is in equilibrium
with a heat bath which could be thought of as
 a black-body radiation in the empty space
surrounding the object.
As a consequence of Eq. (\ref{eq24a}),
different gravitating
configurations of the same size ``in contact" with the same
heat bath may have different surface temperatures.
Therefore, the relevant equilibrium parameter is
the temperature of the heat bath, $T$, and
not the surface temperature of the gravitating object.
 Particle number conservation  yields the constraint
\begin{equation}
\int_0^Rdr\, 4\pi r^2 (1-2{\cal{M}}/r)^{-1/2}\, n(r)=N .
\label{eq45}
\end{equation}
 Given the temperature at infinity  $T$,
the set of self-consistency
equations (\ref{eq40})-(\ref{eq45}) defines
the  general-relativistic Thomas-Fermi equation.
One additional important requirement is that a solution
to the equations (\ref{eq40})-(\ref{eq45})
should  minimize
the free energy.
Based on the considerations given in Appendix \ref{app1},
the free energy may be written
in the form
\begin{equation}
F =M+\mu N -\int dr4\pi r^2 \xi
 (1-2{\cal{M}}/r)^{-1/2}
(p+\rho),
\label{eq27}
\end{equation}
with  $M={\cal M}(R)$.
The theorem proved in Appendix A guarantees that
 solutions to Eqs. (\ref{eq40})-(\ref{eq45})
 extremize the free energy $F$,
i.e., the free-energy functional assumes either
a maximum or a minimum.
We only have to find out which of the solutions are
maxima and discard them as unphysical.

Next we  show  that, in the
Newtonian limit, we recover the
usual Thomas-Fermi model
as defined in Section~\ref{sec2}.
Introducing
the nonrelativistic chemical potential
$\mu_{NR}=\mu-m$
and
 the approximations
$\xi=e^{\varphi}\simeq 1+\varphi$,
$E\simeq m+q^2/2m$
and
  ${\cal M}/r \ll 1$ ,
 we arrive at the Thomas-Fermi self-consistency
 equations \cite{mes,bil1} in the form
\begin{equation}
n=\frac{\rho}{m}
 = g \int \frac{d^3q}{(2\pi)^3}\,
\left(1+\exp(\frac{q^2}{2mT}+\frac{m}{T}\varphi
 -\frac{\mu_{NR}}{T}) \right)^{-1} \, ,
\label{eq46}
\end{equation}
\begin{equation}
\frac{d\varphi}{dr}=\frac{{\cal M}}{r^2} \, ;
\;\;\;\;
\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho \, ,
\label{eq47}
\end{equation}
\begin{equation}
\varphi(R)=-\frac{m N}{R}
\, ; \;\;\;
{\cal{M}}(0)=0,
\label{eq41}
\end{equation}
\begin{equation}
\int_0^R dr\,4\pi r^2 n(r)=N,
\end{equation}
which are equivalent to
 the set of equations (\ref{eq00})-(\ref{eq04}).
The free energy  (\ref{eq27}) in the Newtonian limit yields
\begin{equation}
F=m N +\mu_{NR} N - \frac{1}{2}\int_0^R dr \, 4\pi r^2 n\varphi
-\int_0^R dr \, 4\pi r^2 p \, ,
\label{eq43}
\end{equation}
with
\begin{equation}
 p= g T\int \frac{d^3q}{(2\pi)^3}\,
\ln\left(1+\exp(-\frac{q^2}{2mT}-\frac{m}{T}\varphi
 +\frac{\mu_{NR}}{T}) \right) \, ,
\label{eq48}
\end{equation}
which, up to a constant, is equal the nonrelativistic Thomas-Fermi free
energy (\ref{eq06}).

A straightforward thermodynamic limit
$N\rightarrow\infty$,
as discussed by
Hertel, Thirring, and Narnhofer \cite{her1,her2},
is not directly applicable
 in the general-relativistic case.
 First,
 in contrast to the Newtonian case,
 there exists
  a limiting configuration
with maximal $M$ and $N$ (the OV limit)
at zero temperature,
 and, as we shall shortly demonstrate,
also at finite temperature.
Second,
the scaling property of the relativistic
Thomas-Fermi equation,
which will be referred to as
 {\em relativistic scaling},
is quite distinct from
 nonrelativistic scaling.
This scaling property may be formulated as follows:
  If the configuration
 $\{\xi(r),{\cal{M}}(r)\}$ is a solution to the self-consistency
 equations (\ref{eq40})-(\ref{eq45}), then the
 configuration
 $\{\tilde{\xi}=\xi(A^{-1}r),
 \tilde{\cal M}=A{\cal{M}}(A^{-1}r);A>0\}$
 is also a solution with
 the rescaled
 fermion number
 $\tilde{N}=A^{3/2}N$,
 radius
 $\tilde{R}=AR$,
 asymptotic temperature
 $\tilde{T}=A^{-1/2}T$,
 and fermion mass
 $\tilde{m}=A^{-1/2}m$.
 The free energy is then rescaled as
 $\tilde{F}=AF$.
 Hence, there exists a thermodynamic limit
 of
 $N^{-2/3}F$,
 with
 $N^{-2/3}R$,
 $N^{1/3}T$,
and  $N^{1/3}m$
 approaching constant values
  when
 $N\rightarrow\infty$.

\section{Numerical integration}
\label{sec4}
In the following we use the units in which
$G=1$.
We choose appropriate length, mass and fermion number scales
$a$, $b$, and $c$  respectively, such that
\begin{equation}
a=b=
\sqrt{\frac{2}{g}}\,
\frac{1}{m^2},
\;\;\;\;
c=
\frac{b}{m}  ,
\label{eq81}
\end{equation}
or,  restoring $\hbar$, $c$, and $G$,
we have
\begin{equation}
a=
\sqrt{\frac{2}{g}} \,
\frac{\hbar M_{\rm{Pl}}}{c m^2}
=1.3185 \times 10^{10}\,
\sqrt{\frac{2}{g}} \,
\left(
\frac{15{\rm keV}}{m}
\right)^2
{\rm km},
\end{equation}
\begin{equation}
b=
\sqrt{\frac{2}{g}} \,
\frac{M_{\rm{Pl}}^3}{m^2}
=0.8929 \times 10^{10}\,
\sqrt{\frac{2}{g}} \,
\left(
\frac{15{\rm keV}}{m}
\right)^2
M_{\odot}
\end{equation}
\begin{equation}
c=
\sqrt{\frac{2}{g}} \,
\frac{M_{\rm{Pl}}^3}{m^3}
=5.5942 \times 10^{71}\,
\sqrt{\frac{2}{g}} \,
\left(
\frac{15{\rm keV}}{m}
\right)^3
\label{eq82}
\end{equation}
where $M_{\rm{Pl}}=\sqrt{\hbar c/G}$
denotes the Planck  and $M_{\odot}$ the solar mass.

We are looking
 for a solution of the Thomas-Fermi problem
as a function of temperature.
For numerical convenience,
let us introduce a new parameter
\begin{equation}
\alpha=\frac{\mu}{T}
\label{eq86}
\end{equation}
and the substitution
\begin{equation}
\xi=\frac{\mu}{m}(\Phi+1)^{-1/2}.
\label{eq87}
\end{equation}
Using this and (\ref{eq81}),
Eqs.~(\ref{eq30})-(\ref{eq32}) may be written
in the form
\begin{equation}
n=\frac{1}{\pi^2}
               \int^{\infty}_{0} dy\,
\frac{y^2}{1+\exp [(\sqrt{(y^2+1)/(\Phi+1)}-1)\alpha]} \, ,
\label{eq83}
\end{equation}
\begin{equation}
\rho=\frac{1}{\pi^2}
               \int^{\infty}_{0}dy\,
\frac{y^2\sqrt{y^2+1}}
{1+\exp [(\sqrt{(y^2+1)/(\Phi+1)}-1)\alpha]} \, ,
\label{eq84}
\end{equation}
\begin{equation}
p=\frac{1}{3\pi^2}
               \int^{\infty}_{0}dy\,\frac{y^4
}{\sqrt{y^2+1}
(1+\exp [(\sqrt{(y^2+1)/(\Phi+1)}-1)\alpha]} \, .
\label{eq85}
\end{equation}
In this way, both the fermion mass and the
 chemical potential are eliminated from the
equation of state.

The field equations  (\ref{eq40}) and (\ref{eq42})   now read
\begin{equation}
\frac{d\Phi}{dr} =
-2(\Phi+1)\frac{{\cal M}+4\pi r^3 p}{r(r-2{\cal{M}})} \, ,
\label{eq88}
\end{equation}
\begin{equation}
\label{eq89}
\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho.
\end{equation}
To these two equations we add
\begin{equation}
\frac{d{\cal N}}{dr}=4\pi r^2 (1-2{\cal{M}}/r)^{-1/2} n
\label{eq93}
\end{equation}
imposing
 the particle-number constraint as
a condition at the boundary
\begin{equation}
{\cal N}(R)=N.
\label{eq94}
\end{equation}
Equations (\ref{eq88})-(\ref{eq93})
should be integrated using
the boundary conditions at the origin
\begin{equation}
\Phi(0)=\Phi_0 > -1
\, ; \;\;\;\;\;
{\cal{M}}(0)=0;
\, ; \;\;\;\;\;
{\cal{N}}(0)=0.
\label{eq90}
\end{equation}
The parameter $\Phi_0$,
which is uniquely related to the central
density and pressure,
will eventually be fixed
by the requirement (\ref{eq94}).
For $r\geq R$, the function $\Phi$ yields
the usual empty-space Schwarzschild
solution
\begin{equation}
\Phi(r)=\frac{\mu^2}{m^2}
\left(1-\frac{2 M}{r}\right)^{-1}-1\, ,
\label{eq91}
\end{equation}
  with
\begin{equation}
M={\cal M}(R)=\int_0^R dr 4\pi r^2 \rho(r) .
\label{eq92}
\end{equation}

We now show that
a solution to
the general-relativistic Thomas-Fermi
equation exists
provided the number of fermions
is smaller than a
certain number $N_{\rm max}$
that depends on
$\alpha$ and $R$.
From  (\ref{eq84}) and
  (\ref{eq85})   it follows that,
for any $\alpha>0$,
the equation of state $\rho(p)$ is an
infinitely smooth function and
$d\rho /dp > 0$ for $p > 0$.
Then, as shown by
Rendall and Schmidt
\cite{ren},
 there exists for any value of
the central density $\rho_0$
a unique static, spherically symmetric solution of
the field equations
with $\rho \rightarrow 0$  as
$r$ tends to infinity.
In that limit
${\cal M}(r)\rightarrow\infty$
and
${\cal N}(r)\rightarrow\infty$,
as may be easily seen
by analyzing
the
$r\rightarrow \infty$
limit of
Eqs. (\ref{eq88}) and (\ref{eq89}).
However, the enclosed mass
$M$ and the number of fermions $N$
within a given radius $R$ will be finite.
We can then cut off the matter
from $R$ to infinity
and join the interior solution onto
the empty space Schwarzschild
exterior solution
by making use of
equation
 (\ref{eq91}).
 This equation together with
 (\ref{eq86})
 fixes the chemical potential
 and the temperature at infinity.
Furthermore, it may be shown that
our equation of state obeys
asymptotically  at high densities
a $\gamma$-law,
i.e.,
$\rho=$ const $n^{\gamma}$
and $p=(\gamma-1) \rho$,
with
$\gamma=4/3$.
In this case,
as is well known
\cite{har},
there exists a limiting configuration
$\{ \psi_{\infty}(r),{\cal{M}}(r)_{\infty}\}$
such that $M$ and $N$
approach nonzero values $M_{\infty}$
and $N_{\infty}$,
respectively,
as the central density
$\rho_{0}$
tends to infinity.
Thus, the quantity
 $N$ is a continuous function of
 $\rho_{0}$
on the interval $0 \leq \rho_0 < \infty$,
with $N=0$ for
 $\rho_{0}=0$,  and
 $N=N_{\infty}$
 as
 $\rho_{0}\rightarrow\infty$.
The range of $N$
depends on
$\alpha$ and $R$
and its
upper bound
 may be denoted by
$N_{\rm max}(R,\alpha)$.
Thus, for given
$\alpha$, $R$
and
$N<N_{\rm max}(R,\alpha)$
the set of self-consistency
equations (\ref{eq83})-(\ref{eq92}) has
at least one solution.

As is evident from the equation of state (\ref{eq30})-(\ref{eq32}),
if we do not fix the boundary and do not
constrain the particle number $N$, the
pressure (and the density) will never vanish
(except perhaps at $r=\infty$),
unless $T=0$. Thus, since we fix the boundary at $R$
and cut off the matter from $R$ to infinity,
the pressure (and the density) will have a discontinuity.
This characteristic of the non-relativistic Thomas-Fermi model
in atomic physics \cite{fay},
and Newtonian gravity \cite{her1,mes,bil1}
remains also in general relativity.
However, the density and the pressure decrease rapidly
with $r$, so if $R$ is chosen sufficiently
large, the pressure and the density at the boundary
will be extremely small.
Furthermore, the region $r>R$ is
never empty in reality, so that a positive pressure at the
boundary is more realistic than
a vanishing pressure.

The numerical procedure is now straightforward.
For a fixed, arbitrarily chosen
$\alpha$,
 we first  integrate
equations
 (\ref{eq88}) and (\ref{eq89})
 numerically
on the interval $(0,R)$ and find
solutions
 for various initial
$\Phi_0$.
Simultaneously integrating (\ref{eq93}),
we obtain
${\cal N}(R)$ as a function of $\Phi_0$.
The specific value of $\Phi_0$
is then determined
such that
${\cal N}(R)=N$.
The chemical potential $\mu$
corresponding to
this particular solution
 is given by
(\ref{eq91}).
If we now eliminate $\mu$
using (\ref{eq86}), we finally get the
parametric dependence on temperature
through $\alpha$.

Let us first discuss
a degenerate fermion gas ($T=0$)
 as a reference point
 that can also be compared with the
 well-known results by Oppenheimer and Volkoff
 \cite{opp}.
In this case,
the Fermi distribution in
(\ref{eq83})-(\ref{eq85})
becomes a step function that  yields
an elementary integral with the upper limit
$y_F=\sqrt{\Phi}$ related to the Fermi
momentum
\begin{equation}
q_F=m\sqrt{\Phi}.
\label{eq95}
\end{equation}
The equation of state can be expressed
in terms of elementary functions
of $\Phi$
\begin{equation}
n=\frac{1}{3\pi^2}\Phi\sqrt{\Phi},
\label{eq96}
\end{equation}
\begin{equation}
\rho=\frac{1}{8\pi^2} \left[(2\Phi+1)\sqrt{\Phi(\Phi+1)}
-{\rm Arsh}\sqrt{\Phi}\right] ,
\label{eq97}
\end{equation}
\begin{equation}
p=\frac{1}{24\pi^2} \left[(2\Phi-3)\sqrt{\Phi(\Phi+1)}
+3{\rm Arsh}\sqrt{\Phi}\right] .
\label{eq98}
\end{equation}
The radius of the star is naturally defined as
the point where the density vanishes.
At this point, owing to (\ref{eq96}),
$\Phi=0$.
Therefore, we integrate equations
(\ref{eq83})-(\ref{eq85})
starting from $r=0$ up to the point $R$
where $\Phi(R)=0$.
As a result, the quantities $M$, $N$, and $R$
are obtained
as functions of the parameter $\Phi_0$,
which is related to the central particle-number
  density through (\ref{eq96}).
In Fig.~\ref{fig1}, we plot the mass of the star as
a function of the radius $R$. The maximum of the curve
corresponds to the
OV limit \cite{opp}.
The limiting values are
$R_{\rm OV}=3.357$,
$M_{\rm OV}=0.38426$, and
$N_{\rm OV}=0.39853$,
in units of $a$, $b$, and $b/m$ respectively.
The curve
left from the maximum  represents unstable configurations
that curl up around the point corresponding to the
infinite central density limit.

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig1.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Mass versus radius for fermion stars at
zero temperature
in the general-relativistic framework
 (solid line)
 compared with the corresponding
Newtonian approximation (dotted line).
Dashed line is the black hole limit $M=R/2$.}
\label{fig1}
\end{figure}

We now  turn to the study of nonzero temperature.
The quantities
$T$, $N$, and, $R$ are free parameters in our model
and their
range and choice are dictated
by physics.
The temperature $T$ is restricted only to positive values.
The number of fermions $N$
 is restricted by the Oppenheimer-Volkoff limit.
 The radius
$R$ is theoretically unlimited; practically,
it should not exceed the order of interstellar distances.
It is known that a classical,
semidegenerate, isothermal configuration has no natural boundary
in contrast to the degenerate
 case of zero temperature,  where
for given $N$ (up to the Oppenheimer-Volkoff limit)
the radius $R$ is naturally fixed by the condition of
vanishing pressure and density.
At nonzero  temperature,
if we, e.g., fix only $N$ and $T$
and let $R\rightarrow \infty$, our gas will  occupy
the entire space, and hence $p$ and $\rho$ will
vanish everywhere.
If we do not restrict $N$ and integrate the
equations  on the interval $(0,\infty)$, $M$ and $N$ will
diverge at $\infty$.
In such a case one has to introduce a cutoff.
In the isothermal model of a similar kind
by Chau, Lake, and Stone \cite{cha},
a cutoff was chosen
at the radius $R$, where the energy density
was about six orders smaller than the central value.
Our choice is based on the following
considerations:
As in the Newtonian Thomas-Fermi model
\cite{bil1},
we expect that,
for given
number of fermions
 $N <  N_{\rm OV}$,
there exists a unique
configuration
that is  a solution
to the self-consistency equations
(\ref{eq30}) to (\ref{eq45})
and which
 becomes
a degenerate Fermi gas
at $T=0$.
For such a configuration,
an effective radius
$R_{\rm eff}\ge R_{\rm OV}$
may be defined so that
  $\Phi(R_{\rm eff})=0$.
 Although the density does not vanish at this point,
most of the mass will be contained inside the sphere
 of radius $R_{\rm eff}$.
 If we choose the boundary at
 $R\gg R_{\rm OV}$,
 the total mass will be dominated by the
 density distribution within $R_{\rm eff}$,
 and it will be almost independent of the
 choice of $R$.
 Thus,
 in the following we will work with
 $R=100\simeq 30 R_{\rm OV}$.

In Fig.~\ref{fig2}, the fermion number $N$ is plotted as a function of
initial $\Phi_0$ for several values of the parameter
$\alpha$.
In contrast to the $T=0$ case, all curves with
finite $\alpha$ have a peak around
$\Phi=0$.
The second peak corresponds to the OV
limit.
From this figure we can deduce that, for a given
$N$, there is a range of $\alpha$'s
for which the Thomas-Fermi equation may have
more than one solution.
This is a clear indication for
the existence of an
instability even below the
OV limit
and, as a consequence, we expect that
 a first-order phase transition occurs.

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig2.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Fermion number $N$versus central potential $\Phi_0$
for $\alpha=\frac{\mu}{T}=300$ (full line)
500 (dashed line)
and 150 (dot-dashed line).
T=0 is represented by dotted line.}
\label{fig2}
\end{figure}

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig3.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Temperature T (in units of $m$) versus $1/\alpha$
for $N=0.38$ and $R=100$.}
\label{fig3}
\end{figure}

Fixing $N=0.38$, which is slightly below
the OV limit, we can now plot the temperature as a
function of $\alpha$ in Fig.~\ref{fig3}.
Using this figure as a parametric function for temperature,
 the mass, free energy, and entropy
are shown as
functions of temperature in Figs.
\ref{fig4},
\ref{fig5},
and
\ref{fig6}, respectively.
In the temperature interval
 $T=(0.0015-0.007) m$
there are
three distinct solutions of which
 only two are physical,
 namely, those
 for which the free energy
  assumes a minimum.
 The solution
 that can be continuously extended to any temperature
 above the mentioned interval
 is referred to
 as ``gas", whereas the solution that
  continues to exist at  low
 temperatures, and eventually becomes a degenerate
 Fermi gas at $T=0$, will be called ``condensate".
 In Fig.~\ref{fig2} the gas is represented
 on each curve by the
 part left from the first maximum,
 while
 the part from
 the first minimum up to the second maximum
 represents the condensate.
 By noting that $\Phi_0$ is negative
 for the gas and positive for the condensate,
 we may define an order parameter as
 \begin{equation}
 \delta=\Phi_0+|\Phi_0| ,
 \label{eq99}
 \end{equation}
 which is strictly positive in
 the condensed phase (ordered phase)
 and equal to zero in the gaseous phase
 (disordered phase).

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig4.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Total mass (in units of $b$) versus temperature
for $N=0.38$ and $R=100$.}
\label{fig4}
\end{figure}

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig5.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Free energy per fermion $F/N$
(in units of $m$)
 versus temperature T
for $N=0.38$ and $R=100$.}
\label{fig5}
\end{figure}

\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig6.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Entropy per fermion $S/N$ versus temperature $T$
for $N=0.38$ and $R=100$.}
\label{fig6}
\end{figure}

The phase transition takes place at the
temperature $T_c$,
where the free energy of the gas and
that of the condensate
become equal.
The dashed curves in Figs.
\ref{fig4},
\ref{fig5},
and
\ref{fig6} represent the physically unstable solution.
In our example,
the transition temperature is
$T_c=0.0043951 m$,
as indicated in the plots by the dotted line.
The latent heat per fermion released
during the phase transition is given by the
mass difference at the point of
discontinuity
\begin{equation}
\frac{\Delta M}{N}=0.0438 m .
\end{equation}

So far, we have studied, as an example, an object with
number of fermions $N$ just below the OV limit
$N_{\rm OV}$.
 Any  object with $N<N_{\rm OV}$
will undergo a gravitational phase transition
at a
critical temperature which depends on the mass, of course.
With decreasing $N$,
the cavity  radius $R$
must be appropriately increased, since
the effective radius of the condensate increases
following approximately the
 zero-temperature
mass-radius relation.
As $N$ becomes smaller,
the system approaches
the nonrelativistic scaling regime
discussed in Sec.  \ref{sec2},
and for
$N\ll N_{\rm OV}$
the critical temperature will decrease according to
\begin{equation}
T_c={\rm const}\, N^{4/3},
\end{equation}
if the cavity  radius $R$ is simultaneously rescaled as $N^{-1/3}$.
In Fig.~{\ref{fig7}
we compare the critical temperature calculated
in both Newtonian and general-relativistic Thomas-Fermi models,
as a function of $N$.
The nonrelativistic scaling law turns out to be very accurate
for $N<0.2\,N_{\rm OV}$.
\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig7.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Critical temperature as a function
of the fermion number
in the Newtonian (dashed line) and general-relativistic
Thomas-Fermi model (solid line).}
\label{fig7}
\end{figure}

It is important to check that the critical
temperature is not very sensitive to
 variations of the cavity radius $R$
 for the following two reasons:
First, in our model $R$ is arbitrary except for the
requirement that it should be much larger then the
effective radius which
for
 $N=0.38$ is of the order
$R_{\rm eff}\simeq R_{\rm OV}=3.357$.
Second,
if the critical temperature rapidly  decreases with
$R$, the adiabatic cooling of the gas through the
universe expansion may not necessarily lead to
the point of the phase transition.
 Fig.~\ref{fig8} shows that the critical
temperature indeed decreases very slowly,
roughly by a factor of two if $R$ increases
from 30 to 300.
This is much weaker than the adiabatic cooling
of a nonrelativistic gas
which goes approximately as $1/R^2$.
Thus we conclude that the gravitational phase transition
will necessarily take place in the course of
the universe expansion.
\begin{figure}
\vbox{\vskip230pt
\special{psfile=fig8.ps hoffset=80 voffset=-80 vsize=230
hscale=50 vscale=50}}
\caption{Critical temperature $T_c$ as a function
of the cavity radius $R$
for $N=0.38$.}
\label{fig8}
\end{figure}

\section{Conclusions}
\label{sec6}

In this work,
we have extended the Thomas-Fermi model
to a general-relativistic
 framework.
 This model was
 then applied
to  a system of self-gravitating
 fermions.
 We have
  investigated numerically
the circumstances under which
this system undergoes
a gravitational phase transition
 at nonzero temperature.
This phase transition is quite distinct from
the more extensively investigated strong-interaction driven phase
transition that might occur in neutron stars
\cite{saw,wit}.
The main underlying physics here
is the competition between the
partial degeneracy pressure due to the
Fermi-Dirac statistics and the
attractive force due to the gravitational interaction.
 It is obvious that the application of this model to
 astrophysical systems
 will work very well if the non-gravitational interactions
 between the individual particles can be neglected.
This is certainly the case, e.g., for
weakly interacting quasidegenerate heavy neutrino
neutralino, or gravitino matter \cite{bil2,bil1,cha,kul,bil3},
but perhaps it could be valid even for collisionless stellar systems
\cite{shu,chav}.

Finally, let us briefly comment on a
similar model by
Chau, Lake and Stone \cite{cha} which was considered
earlier
in the context of a possible galactic massive neutrino halo.
Their model differs  from our approach in essentially two aspects:
First, the equation of state is not consistent with
the condition of thermodynamical and chemical
equilibrium, i.e. with Eq.~(\ref{eq24a})
and second,
the particle number constraint (\ref{eq45})
is not imposed.
Thus, in contrast to the Thomas-Fermi model
discussed here,
the Chau et al.
 model does not describe a
canonical system in
equilibrium.

\subsection*{Acknowledgment}
We acknowledge useful discussions with
 D. Tsiklauri.
 This work  was supported by
 the Foundation for Fundamental Research
 (FFR).