have very similar present-day spatial distributions compared to the *in-situ* stars.

According to the E-Lz definition of Belokurov & Kravtsov (2023, 2024), both the Kraken globular clusters and the Heracles stars are considered part of Aurora, hence born *in-situ* – this was also argued by Myeong et al. (2022) based on similarity between Heracles and Aurora in various chemical spaces. Horta et al. (2023b) show that there is a statistical difference between the detailed chemistry of the Heracles building block and Aurora stars – this might partly be due to the Heracles stars being selected to have specific chemistry (low [Al/Fe] and high [Mg/Mn]), but the authors argue that their results are still an indication that the Heracles stars are of distinct origin from the *in-situ* stars.

Furthermore, the contribution from Gaia-Sausage/Enceladus (GS/E; Belokurov et al. 2018; Helmi et al. 2018), which was accreted more recently (~10 Gyr ago), is expected to be small in the inner few kpc of our Galaxy. This is especially true among confined stars, since GS/E stars typically have low energies (above the orange line in Fig. 13) and very radial orbits with large apacentres. But if any GS/E debris is present/confined in the inner Galaxy, it is more likely to be metal-rich since those stars are expected to have been most tightly bound to the GS/E progenitor and therefore sunk deepest in the MW potential (e.g. Amarante et al. 2022; Orkney et al. 2023). GS/E is therefore unlikely a significant contributor to the metal-poor inner Galaxy.

**Small building blocks** – The expectation is that smaller dwarf galaxies should have contributed to the primordial Milky Way as well, although their total mass contribution is expected to be minor according to e.g. the analysis of the FIRE simulations by Horta et al. (2023a) – they find that a proto-Milky Way is typically dominated by one or two dominant systems and a few smaller ones. Small (and/or very small) building blocks may contribute more in the lower metallicity tail of the population; see e.g. Orkney et al. (2023), who investigated the fraction of *ex-situ*/accreted stars in the Auriga simulations. Focusing on R<sub>GC</sub> < 5 kpc, they find that the fraction of accreted stars (from a range of progenitors) steadily rises with decreasing metallicity. In their main simulation, the accreted fraction is ~35 per cent for [Fe/H] = -1.5 and rises to 60/70 per cent for [Fe/H] < -2.0 and < -2.5, respectively. The accreted fraction of [Fe/H] < -2.5 stars is between 60 and 80 per cent for nine out of their ten Milky Way simulations, only one simulation has a somewhat lower fraction of 40 per cent. These numbers do not significantly change when considering stars that have apacentres within 5 kpc (instead of just their present-day location). The decrease in fraction of tightly confined stars with decreasing metallicity we observe in PIGS (Fig. 7) might be connected to the increase in accreted stellar populations. It is worth noting that our metallicities are not necessarily on the same scale as those in the simulations, but the trends remain.

Finally, as discussed in the introduction, it has also been shown that the fraction of stars with globular cluster-like chemistry (based on their high nitrogen abundances) is larger in the inner Galaxy (Schiavon et al. 2017; Horta et al. 2021b; Belokurov & Kravtsov 2023). The latter authors argue that these mostly come from disrupted *in-situ* globular clusters – and are therefore part of the same population as Aurora. All these studies are based on APOGEE data, typically for metallicities of [Fe/H] ≳ -1.7, and it is still an open question what the contribution of disrupted globular clusters might be for VMP stars. Some VMP stars in the inner Galaxy do appear to come from globular clusters; Sestito et al. (2023) found two stars with [Fe/H] < -2.5 that show the typical globular cluster signature of enhanced [Na/Mg]. Arentsen et al. (2021) discussed the possibility

that the apparently low fraction of carbon-enhanced very metal-poor s-process enhanced (CEMP-s) stars in the inner Galaxy could be due to a lower VMP binary fraction, which in turn could be due to a larger contribution from disrupted globular clusters (that have low binary fractions). However, further work on the chemistry of inner Galaxy VMP stars is necessary to investigate the role of disrupted VMP globular clusters in this region.

### 5.2.3 Rotational properties of the metal-poor inner Galaxy

Simulations show that proto-galactic populations typically show weak but systematic net rotation up to a few tens of km s⁻¹ (Belokurov & Kravtsov 2022; Chandra et al. 2023; McCluskey et al. 2024; Horta et al. 2024). We indeed find rotation for PIGS metal-poor inner Galaxy stars (Fig. 10). Our results are consistent with previous work that also found some azimuthal velocity among metal-poor inner Galaxy stars and/or local likely *in-situ* stars (Wegg et al. 2019; Kunder et al. 2020; Lucey et al. 2021; Belokurov & Kravtsov 2022; Conroy et al. 2022; Rix et al. 2022), although Rix et al. (2022) claim that there is no net rotation among inner Galaxy stars with [M/H] < -2.0. A net $v_\phi$ of ~+70 km s⁻¹ has also been found for metal-poor ([Fe/H] < -1.0) *in-situ* globular clusters with R<sub>GC</sub> < 5 kpc (Belokurov & Kravtsov 2024), where *in-situ* was defined using the E-Lz Aurora boundary.

A proto-galactic population could have been spinning since its formation (e.g. due to a net angular momentum from the combination of building blocks), but this does not have to be the case – in their simulations, McCluskey et al. (2024) show that the oldest *in-situ* stars were born with low velocity dispersion and no net rotation, but currently this population has the highest dispersion and has been spun up (and flattened) over time (see also Chandra et al. 2023). The mechanism(s) for the spinning up of the primordial Milky Way are still under discussion – the growing disc likely plays a role, as well as the Galactic bar, which can severely affect the orbits of stars, moving them onto preferentially prograde orbits (see e.g. Pérez-Villegas, Portail & Gerhard 2017; Dillamore et al. 2023). Preferentially prograde merger events could also add to the spin of the ancient inner Galaxy population.

In our PIGS sample, we find that the azimuthal velocity decreases as a function of metallicity for stars with 1.5 < R<sub>GC</sub> < 3.5 kpc between -2.0 < [Fe/H] < -1.0 (Fig. 10), even after removing halo interlopers with large apacentres. It remains roughly constant for [Fe/H] < -2.0, with $v_\phi$ ~ 40 km s⁻¹. We also find that $v_\phi$ stays constant with metallicity at a similar $v_\phi$ closer to the Galactic centre (R<sub>GC</sub> < 1.5 kpc), across all metallicities (possibly with a slight drop for [Fe/H] < -2.0). One effect to keep in mind is that the more metal-rich stars in our sample are typically slightly further away from the Galactic centre than the more metal-poor stars (see Fig. 5) even within our selected rings of R<sub>GC</sub>, which might allow them to have slightly higher net azimuthal velocities. A rise in rotation is also seen in the metal-rich APOGEE data from Q21, which has higher $v_\phi$ further away from the Galactic centre – this might be related to the rise in the rotation curve of the Galaxy (although in APOGEE it might also be connected to a rising contribution from the disc/bar).

In previous work with the PIGS data (Arentsen et al. 2020a), the rotation curve was studied as a function of metallicity by projecting the radial velocities with respect to the Galactic centre. They observed that there was no significant rotational signature for stars with [Fe/H] < -2.0. This difference with the results in this work may be due to several factors: the previous work did not do a cut on apacentre and therefore included many ‘halo interlopers’ (which
have small azimuthal velocities), the sample of VMP stars was only half the size (and less well-distributed in *l* and *b*), and there is limited sensitivity when only using projected radial velocities.

Additionally, Rix et al. (2022) claim to find no net rotation among inner Galaxy VMP stars, looking at $\bar{J}_{\phi}/J_{\text{tot}}$ (see their fig. 6). However, compared to PIGS, their analysis is limited to stars with slightly larger apocentres (3–7 kpc, partly because their kinematics sample has larger distances from the Galactic centre), and their figure does not actually show a drop to completely zero rotational support for [M/H] < −2.0. Their results are therefore not in conflict with ours.

### 5.2.4 Interpretation

There are currently no predictions for velocity trends with metallicity within proto-galactic populations to compare to. We can, however, speculate about the origin for the trends in the PIGS data. Previously, Arentsen et al. (2020a) interpreted the trend of decreasing inner Galactic rotation and increasing velocity dispersion as a function of metallicity in PIGS (see also the small symbols in Fig. 11) as the result of a density transition between two overlapping populations: the metal-rich, cylindrically rotating bulge that formed from the disc and a hotter metal-poor halo component with no net rotation. However, the cylindrically rotating bulge is shown to be predominantly made up of stars with [Fe/H] > −1.0 with no strong tail towards lower metallicities, and stars with [Fe/H] < −1.0 are usually interpreted as coming from the halo or a spheroidal bulge component (e.g. Q21; Wylie et al. 2021; Razera et al. 2022). This is consistent with recent works suggesting the MW thick disc does not start to form until [Fe/H] ~ −1.3 and only becomes prominent above [Fe/H] = −1.0 (e.g. Belokurov & Kravtsov 2022; Chandra et al. 2023). It therefore appears unlikely that the $v_{\phi}$ trend with [Fe/H] is (entirely) due to a transition between metal-poor halo and metal-rich discy bulge.

Recently, Zhang, Ardern-Arentsen & Belokurov (2023) used the Andrae et al. (2023) metallicities and kinematics from Gaia to fit Gaussian Mixture Models to the velocity distributions of Solar neighbourhood stars. They showed that there appear to be two different halo components (separate from GS/E) in the metal-poor and very metal-poor regime – a stationary halo with $\bar{v}_{\phi} \approx 0$ km s⁻¹ and $\sigma_{v_{\phi}} \approx 130$ km s⁻¹ and a prograde halo with $\bar{v}_{\phi} \approx +70$ km s⁻¹ and $\sigma_{v_{\phi}} \approx 90$ km s⁻¹ (which they suggest might be related to Aurora). They find the stationary halo to be dominant for [M/H] < −2.0 (although the prograde halo is still present), and for −2.0 < [M/H] < −1.0 the prograde halo takes over while the contribution from the stationary halo drops strongly (and the disc becomes a relevant component for [M/H] > −1.3).

If these two halo components are also present in the inner Galaxy, as suggested by our detection of two components in the IMP regime with similar mean and dispersion in $v_{\phi}$ (Section 4.3 and Table 2) compared to the two inner halo components in Zhang et al. (2023), we might interpret the trends we see between $v_{\phi}$, $\sigma_{v_{\phi}}$ and [Fe/H] for $1.5 < R_{GC} < 3.5$ kpc as a change in contributions from the stationary and prograde halos. The fraction of the cooler and faster component in our analysis varies from 0.83 to 0.62 to 0.40 for the MP, IMP, and VMP regimes, respectively. Connecting it to the rest of the discussion here as well as that in Zhang et al. (2023), this suggests a transition from predominantly ‘accreted’ stars (or smaller building blocks) at lower metallicity to predominantly ‘in-situ’ stars (or larger building blocks) at higher metallicities. We find that our slower halo component still has a slight rotation ($v_{\phi} = 26$ km s⁻¹, as opposed to the ‘stationary’ component by Zhang et al. (2023). Our estimate is for stars with $1.5 < R_{GC} < 3.5$ kpc, while Zhang

et al. (2023) focus on stars closer to the Solar neighbourhood – it is possible that this component rotates faster closer to the Galactic centre. Along the same line of reasoning, the population of stars even closer to the Galactic centre ($0.5 < R_{GC} < 1.5$ kpc, for which $\bar{v}_{\phi} \approx +45$ km s⁻¹ independent of metallicity, may be dominated by the in-situ population at all metallicities.

## 5.3 Future work

There is limited information currently available for the metal-poor inner Galaxy. In this work we employed spectroscopic metallicities and dynamical properties for thousands of metal-poor inner Galaxy stars, which have been observed with low/medium-resolution spectroscopy. Improvements in both the chemistry and dynamics could be made with higher resolution spectroscopic observations.

The dynamics will be improved thanks to the improved proper motions in the next Gaia data release, but in particular the dynamics would improve if it were possible to get more precise distances. One limitation is the parallax quality, which will only get slightly better in the next Gaia data release. On the other hand, higher resolution spectra would allow for more precise stellar parameter estimates, which would improve the STARHORSE distance estimates. In addition, more work could go into investigating how appropriate the adopted isochrones are for VMP stars (as well as e.g. HB stars). Finally, there is still much to learn about (inhomogeneous) extinction towards the Galactic bulge, which also impacts distance estimates that include photometry.

Higher resolution spectroscopic follow-up also allows for the measurement of elemental abundances other than just ‘metallicity’ (and carbon). The current samples of high-resolution observations in the inner Galaxy are relatively small and/or inhomogeneous, although we have already learned a lot from them (see the Introduction). Large, homogeneous samples of metal-poor inner Galaxy stars will be observed with the upcoming Southern hemisphere 4-m Multi-Object Spectroscopic Telescope (4MOST; de Jong et al. 2019). Over 25 000 metal-poor candidates pre-selected from PIGS photometry will be observed in the VMP-PIGS sub-survey (more details in a future paper, Ardern-Arentsen et al. in preparation) within the 4MIDABLE-LR consortium survey (Chiappini et al. 2019), as well as VMP candidates based on metallicities from STARHORSE + Gaia XP spectra. The LR mode in 4MOST is significantly higher resolution ($R \sim 6500$) than what is currently available in terms of ‘low-resolution’ surveys (which are more typically $R \le 2000$). A variety of chemical abundances is expected to be derived from these spectra, among them carbon abundances, individual alpha and iron-peak elements, as well as some s-process elements. This will be the first large sample of thousands of very metal-poor inner Galaxy stars with chemical abundances beyond simple metallicity. There will also be some dedicated follow-up of VMP candidates in the high resolution 4MIDABLE-HR survey (Bensby et al. 2019), for which detailed chemical abundances from all nucleosynthetic channels are expected to be derived. Other ongoing and upcoming surveys will also target the inner Galaxy [e.g. SDSS-V, Kollmeier et al. (2019) and MOONS, Cirasuolo et al. (2011)], but they are not necessarily targeting VMP stars specifically, and/or focusing on the infrared, which has fewer lines for VMP stars.

What are the questions that can be probed with these large samples of homogeneous chemical abundances? Hopefully these large samples of stars with homogeneous chemical abundance will shed light on the origin of inner Galaxy metal-poor stars, e.g. are they more likely to have been born in the main progenitor of the Milky Way, or in a small dwarf galaxy? What were the properties of
have small azimuthal velocities), the sample of VMP stars was only half the size (and less well-distributed in *l* and *b*), and there is limited sensitivity when only using projected radial velocities.

Additionally, Rix et al. (2022) claim to find no net rotation among inner Galaxy VMP stars, looking at $\bar{J}_{\phi}/J_{\text{tot}}$ (see their fig. 6). However, compared to PIGS, their analysis is limited to stars with slightly larger apocentres (3–7 kpc, partly because their kinematics sample has larger distances from the Galactic centre), and their figure does not actually show a drop to completely zero rotational support for $[\text{M/H}] < -2.0$. Their results are therefore not in conflict with ours.

### 5.2.4 Interpretation

There are currently no predictions for velocity trends with metallicity within proto-galactic populations to compare to. We can, however, speculate about the origin for the trends in the PIGS data. Previously, Arentsen et al. (2020a) interpreted the trend of decreasing inner Galactic rotation and increasing velocity dispersion as a function of metallicity in PIGS (see also the small symbols in Fig. 11) as the result of a density transition between two overlapping populations: the metal-rich, cylindrically rotating bulge that formed from the disc and a hotter metal-poor halo component with no net rotation. However, the cylindrically rotating bulge is shown to be predominantly made up of stars with $[\text{Fe/H}] > -1.0$ with no strong tail towards lower metallicities, and stars with $[\text{Fe/H}] < -1.0$ are usually interpreted as coming from the halo or a spheroidal bulge component (e.g. Q21; Wylie et al. 2021; Razera et al. 2022). This is consistent with recent works suggesting the MW thick disc does not start to form until $[\text{Fe/H}] \sim -1.3$ and only becomes prominent above $[\text{Fe/H}] = -1.0$ (e.g. Belokurov & Kravtsov 2022; Chandra et al. 2023). It therefore appears unlikely that the $v_{\phi}$ trend with $[\text{Fe/H}]$ is (entirely) due to a transition between metal-poor halo and metal-rich discy bulge.

Recently, Zhang, Ardern-Arentsen & Belokurov (2023) used the Andrae et al. (2023) metallicities and kinematics from Gaia to fit Gaussian Mixture Models to the velocity distributions of Solar neighbourhood stars. They showed that there appear to be two different halo components (separate from GS/E) in the metal-poor and very metal-poor regime – a stationary halo with $\bar{v}_{\phi} \approx 0$ km s⁻¹ and $\sigma_{v_{\phi}} \approx 130$ km s⁻¹ and a prograde halo with $\bar{v}_{\phi} \approx +70$ km s⁻¹ and $\sigma_{v_{\phi}} \approx 90$ km s⁻¹ (which they suggest might be related to Aurora). They find the stationary halo to be dominant for $[\text{M/H}] < -2.0$ (although the prograde halo is still present), and for $-2.0 < [\text{M/H}] < -1.0$ the prograde halo takes over while the contribution from the stationary halo drops strongly (and the disc becomes a relevant component for $[\text{M/H}] > -1.3$).

If these two halo components are also present in the inner Galaxy, as suggested by our detection of two components in the IMP regime with similar mean and dispersion in $v_{\phi}$ (Section 4.3 and Table 2) compared to the two inner halo components in Zhang et al. (2023), we might interpret the trends we see between $v_{\phi}$, $\sigma_{v_{\phi}}$ and $[\text{Fe/H}]$ for $1.5 < R_{\text{GC}} < 3.5$ kpc as a change in contributions from the stationary and prograde halos. The fraction of the cooler and faster component in our analysis varies from 0.83 to 0.62 to 0.40 for the MP, IMP, and VMP regimes, respectively. Connecting it to the rest of the discussion here as well as that in Zhang et al. (2023), this suggests a transition from predominantly ‘accreted’ stars (or smaller building blocks) at lower metallicity to predominantly ‘in-situ’ stars (or larger building blocks) at higher metallicities. We find that our slower halo component still has a slight rotation ($v_{\phi} = 26$ km s⁻¹, as opposed to the ‘stationary’ component by Zhang et al. (2023). Our estimate is for stars with $1.5 < R_{\text{GC}} < 3.5$ kpc, while Zhang

et al. (2023) focus on stars closer to the Solar neighbourhood – it is possible that this component rotates faster closer to the Galactic centre. Along the same line of reasoning, the population of stars even closer to the Galactic centre ($0.5 < R_{\text{GC}} < 1.5$ kpc, for which $\bar{v}_{\phi} \approx +45$ km s⁻¹ independent of metallicity, may be dominated by the in-situ population at all metallicities.

## 5.3 Future work

There is limited information currently available for the metal-poor inner Galaxy. In this work we employed spectroscopic metallicities and dynamical properties for thousands of metal-poor inner Galaxy stars, which have been observed with low/medium-resolution spectroscopy. Improvements in both the chemistry and dynamics could be made with higher resolution spectroscopic observations.

The dynamics will be improved thanks to the improved proper motions in the next Gaia data release, but in particular the dynamics would improve if it were possible to get more precise distances. One limitation is the parallax quality, which will only get slightly better in the next Gaia data release. On the other hand, higher resolution spectra would allow for more precise stellar parameter estimates, which would improve the STARHORSE distance estimates. In addition, more work could go into investigating how appropriate the adopted isochrones are for VMP stars (as well as e.g. HB stars). Finally, there is still much to learn about (inhomogeneous) extinction towards the Galactic bulge, which also impacts distance estimates that include photometry.

Higher resolution spectroscopic follow-up also allows for the measurement of elemental abundances other than just ‘metallicity’ (and carbon). The current samples of high-resolution observations in the inner Galaxy are relatively small and/or inhomogeneous, although we have already learned a lot from them (see the Introduction). Large, homogeneous samples of metal-poor inner Galaxy stars will be observed with the upcoming Southern hemisphere 4-m Multi-Object Spectroscopic Telescope (4MOST; de Jong et al. 2019). Over 25 000 metal-poor candidates pre-selected from PIGS photometry will be observed in the VMP-PIGS sub-survey (more details in a future paper, Ardern-Arentsen et al. in preparation) within the 4MIDABLE-LR consortium survey (Chiappini et al. 2019), as well as VMP candidates based on metallicities from STARHORSE + Gaia XP spectra. The LR mode in 4MOST is significantly higher resolution ($R \sim 6500$) than what is currently available in terms of ‘low-resolution’ surveys (which are more typically $R \le 2000$). A variety of chemical abundances is expected to be derived from these spectra, among them carbon abundances, individual alpha and iron-peak elements, as well as some s-process elements. This will be the first large sample of thousands of very metal-poor inner Galaxy stars with chemical abundances beyond simple metallicity. There will also be some dedicated follow-up of VMP candidates in the high resolution 4MIDABLE-HR survey (Bensby et al. 2019), for which detailed chemical abundances from all nucleosynthetic channels are expected to be derived. Other ongoing and upcoming surveys will also target the inner Galaxy [e.g. SDSS-V, Kollmeier et al. (2019) and MOONS, Cirasuolo et al. (2011)], but they are not necessarily targeting VMP stars specifically, and/or focusing on the infrared, which has fewer lines for VMP stars.

What are the questions that can be probed with these large samples of homogeneous chemical abundances? Hopefully these large samples of stars with homogeneous chemical abundance will shed light on the origin of inner Galaxy metal-poor stars, e.g. are they more likely to have been born in the main progenitor of the Milky Way, or in a small dwarf galaxy? What were the properties of
the First Stars in different environments, and how did they affect the
next stellar generations? What kind of CEMP stars are present in the
inner Galaxy? What is the contribution of disrupted globular cluster
stars in the VMP regime? Additionally, in such a large homogeneous
data set, there is also ample opportunity for unexpected discoveries.

Finally, to be able to compare our dynamics results to simulations,
the right approach would be to forward model those to take into ac-
count the effect of large parameter uncertainties in the observations.
This is beyond the scope of the current work, but is certainly worth
revisiting in the future.

# 6 SUMMARY AND CONCLUSIONS

In this work, we investigated the dynamical properties of (very)
metal-poor stars in the inner Galaxy, using spectroscopic data from
PIGS (Arentsen et al. 2020b). Metal-poor stars in the central regions
of the Milky Way are expected to be among the oldest in our Galaxy.
We derived spectro-photometric distances for the stars in PIGS using
STARHORSE (Santiago et al. 2016; Queiroz et al. 2018), which have
typical uncertainties between 10 – 20 per cent (Fig. 3). Combining
these with radial velocities from PIGS and proper motions from Gaia,
we derive orbital properties by integrating the stars in the Portail et al.
(2017)/S22 potential, which includes a realistic representation of the
Galactic bar. After spectroscopic and photometric quality cuts, there
are ~7500 stars in the final sample used for our dynamical analysis,
of which ~1700 have [Fe/H] < -2.0 (VMP stars). This is the largest
sample of VMP stars with detailed dynamical properties in the inner
Galaxy.

With this paper, we release all the PIGS spectroscopic data
products (see Appendix D). This includes the stellar parameters from
two independent methods and the radial velocities, for which the
methods are described in Arentsen et al. (2020b), and the dynamical
properties inferred in this work, if available. The stars in PIGS are
mostly inner Galaxy stars, but there is also a significant number
(~800) of Sagittarius dwarf galaxy stars in the release (see Vitali
et al. 2022) – although the distances and dynamics in this work
should not be used for Sagittarius stars. The data release of the PIGS
photometry will be part of a future paper (Ardern-Arentsen et al., in
preparation).

Our main findings in this paper are as follows:

(i) Almost all stars in our sample are currently located within 3.5 kpc of the Galactic centre (Fig. 5). We find that VMP stars are found more symmetrically distributed around the Galactic centre than less metal-poor stars (−2.0 < [Fe/H] < −1.0), which are systematically closer to us. This might be influenced by selection effects in our sample (e.g. VMP stars are intrinsically brighter).

(ii) We compare trends in the dynamical properties ($v_\phi$, velocity dispersion, $z_{max}$, apocentres, eccentricities) of metal-poor stars in PIGS ([Fe/H] < -1.5) to metal-rich inner Galaxy stars from APOGEE ([Fe/H] > 0, from Q21), see Fig. 6. The dynamics of metal-rich stars are dominated by the bar, whereas the metal-poor stars do not clearly follow the same structure, and behave more like a spheroidal, pressure-supported population.

(iii) We find that the majority of metal-poor stars in PIGS have orbits that are confined to within ~5 kpc of the Galactic centre (Fig. 7). The fraction of confined stars decreases with decreasing metallicity, but even VMP stars have a significant fraction of stars that is confined to the inner Galaxy (~40 per cent within 3.5 kpc, ~60 per cent within 5 kpc and ~80 per cent within 10 kpc). Given the different selection functions, sample sizes and adoption of different potentials, definitions and distance estimates, previous

estimates in the literature are in reasonable agreement with our
results. We conclude that the Milky Way is hosting a significant,
centrally concentrated, (very) metal-poor Galactic component (in
agreement with e.g. Rix et al. 2022).

(iv) We find that metal-poor stars in the inner Galaxy show coherent rotational motion around the Galactic centre (Fig. 10), down to the very metal-poor regime. For a ring in Galactocentric radius between 1.5 and 3.5 kpc, we find that $\bar{v}_\phi$ changes roughly linearly from ~+80 km s⁻¹ to ~+40 km s⁻¹ between [Fe/H] = -1.0 and -2.2, respectively. At lower metallicity, $\bar{v}_\phi$ remains roughly constant. For stars within a Galactocentric radius of 1.5 kpc, we find $\bar{v}_\phi \approx 45$ km s⁻¹, independent of metallicity, with a slight decrease below [Fe/H] < -2.0. We suggest these trends may be the result of a transition between two spheroidal populations, a more metal-rich one that rotates faster (possibly connected to stars formed in-situ/in large building blocks) and a more metal-poor one that rotates less fast or not at all (possibly connected to stars that were accreted/formed in smaller building blocks). This is supported by a preferred two-component fit to the velocity distribution for stars with -2.0 < [Fe/H] < -1.5, with parameters similar to the two halo components recently found by Zhang et al. (2023).

(v) The high velocity dispersion of metal-poor inner Galaxy stars is mostly driven by ‘halo interlopers’, and previous PIGS results of a strongly rising velocity dispersion with decreasing metallicity (Arentsen et al. 2020a) can be explained by this (see Fig. 11, consistent with previous findings by Lucey et al. 2021 and Kunder et al. 2020). After removing stars with apocentres larger than 5 kpc, the velocity dispersion is still rising with decreasing metallicity, but only gently: from ~80 km s⁻¹ for [Fe/H] = -1 to ~95 km s⁻¹ for [Fe/H] < -2.0. This is likely due to the VMP inner Galaxy stars having a slightly more extended distribution compared to less metal-poor stars.

(vi) We investigated the orbital properties of PIGS CEMP stars in the inner Galaxy (Arentsen et al. 2021), and found no significant difference between the apocentre distributions of carbon-rich and carbon-normal VMP stars (Fig. 12). More work is necessary to further investigate whether the occurrence of CEMP stars is different between ‘true inner Galaxy stars’ and ‘halo interlopers’.

The past years have seen significant progress in the study of the metal-poor inner Galaxy, but there are still many open questions about the ancient stellar population(s) in this region. The next years will see many new results from several large spectroscopic surveys targeting the Galactic bulge (e.g. SDSS-V, MOONS, 4MOST), with the 4MOST surveys 4MIDABLE-LR and -HR (Bensby et al. 2019; Chiappini et al. 2019) being particularly promising for the investigation of the metal-poor inner Milky Way. Large, homogeneous samples of metal-poor stars with chemical abundances will allow us to probe the chemical evolution in the earliest phase of our Galaxy.

ACKNOWLEDGEMENTS

We thank the reviewer for their helpful suggestions. AAA thanks
all members of the Pristine collaboration and the Cambridge
Streams group for valuable discussions. We thank Sarah Martell
and Jeffrey Simpson for observing for the PIGS/AAT program.
AAA acknowledges support from the Herchel Smith Fellowship at
the University of Cambridge and a Fitzwilliam College research
fellowship supported by the Isaac Newton Trust. AAA, GM, NFM,
and ZY gratefully acknowledge funding from the European Research
Council (ERC) under the European Unions Horizon 2020 research
and innovation programme (grant agreement No. 834148). GM,
the First Stars in different environments, and how did they affect the
next stellar generations? What kind of CEMP stars are present in the
inner Galaxy? What is the contribution of disrupted globular cluster
stars in the VMP regime? Additionally, in such a large homogeneous
data set, there is also ample opportunity for unexpected discoveries.

Finally, to be able to compare our dynamics results to simulations,
the right approach would be to forward model those to take into ac-
count the effect of large parameter uncertainties in the observations.
This is beyond the scope of the current work, but is certainly worth
revisiting in the future.

6 SUMMARY AND CONCLUSIONS

In this work, we investigated the dynamical properties of (very)
metal-poor stars in the inner Galaxy, using spectroscopic data from
PIGS (Arentsen et al. 2020b). Metal-poor stars in the central regions
of the Milky Way are expected to be among the oldest in our Galaxy.
We derived spectro-photometric distances for the stars in PIGS using
STARHORSE (Santiago et al. 2016; Queiroz et al. 2018), which have
typical uncertainties between 10 – 20 per cent (Fig. 3). Combining
these with radial velocities from PIGS and proper motions from Gaia,
we derive orbital properties by integrating the stars in the Portail et al.
(2017)/S22 potential, which includes a realistic representation of the
Galactic bar. After spectroscopic and photometric quality cuts, there
are ~7500 stars in the final sample used for our dynamical analysis,
of which ~1700 have [Fe/H] < -2.0 (VMP stars). This is the largest
sample of VMP stars with detailed dynamical properties in the inner
Galaxy.

With this paper, we release all the PIGS spectroscopic data
products (see Appendix D). This includes the stellar parameters from
two independent methods and the radial velocities, for which the
methods are described in Arentsen et al. (2020b), and the dynamical
properties inferred in this work, if available. The stars in PIGS are
mostly inner Galaxy stars, but there is also a significant number
(~800) of Sagittarius dwarf galaxy stars in the release (see Vitali
et al. 2022) – although the distances and dynamics in this work
should not be used for Sagittarius stars. The data release of the PIGS
photometry will be part of a future paper (Ardern-Arentsen et al., in
preparation).

Our main findings in this paper are as follows:

(i) Almost all stars in our sample are currently located within 3.5 kpc of the Galactic centre (Fig. 5). We find that VMP stars are found more symmetrically distributed around the Galactic centre than less metal-poor stars (−2.0 < [Fe/H] < −1.0), which are systematically closer to us. This might be influenced by selection effects in our sample (e.g. VMP stars are intrinsically brighter).

(ii) We compare trends in the dynamical properties ($v_\phi$, velocity dispersion, $z_{max}$, apocentres, eccentricities) of metal-poor stars in PIGS ($[Fe/H] < -1.5$) to metal-rich inner Galaxy stars from APOGEE ($[Fe/H] > 0$, from Q21), see Fig. 6. The dynamics of metal-rich stars are dominated by the bar, whereas the metal-poor stars do not clearly follow the same structure, and behave more like a spheroidal, pressure-supported population.

(iii) We find that the majority of metal-poor stars in PIGS have orbits that are confined to within ~5 kpc of the Galactic centre (Fig. 7). The fraction of confined stars decreases with decreasing metallicity, but even VMP stars have a significant fraction of stars that is confined to the inner Galaxy (~40 per cent within 3.5 kpc, ~60 per cent within 5 kpc and ~80 per cent within 10 kpc). Given the different selection functions, sample sizes and adoption of different potentials, definitions and distance estimates, previous

estimates in the literature are in reasonable agreement with our
results. We conclude that the Milky Way is hosting a significant,
centrally concentrated, (very) metal-poor Galactic component (in
agreement with e.g. Rix et al. 2022).

(iv) We find that metal-poor stars in the inner Galaxy show coherent rotational motion around the Galactic centre (Fig. 10), down to the very metal-poor regime. For a ring in Galactocentric radius between 1.5 and 3.5 kpc, we find that $\bar{v}_{\phi}$ changes roughly linearly from ~+80 km s⁻¹ to ~+40 km s⁻¹ between [Fe/H] = -1.0 and -2.2, respectively. At lower metallicity, $\bar{v}_{\phi}$ remains roughly constant. For stars within a Galactocentric radius of 1.5 kpc, we find $\bar{v}_{\phi} \approx 45$ km s⁻¹, independent of metallicity, with a slight decrease below [Fe/H] < -2.0. We suggest these trends may be the result of a transition between two spheroidal populations, a more metal-rich one that rotates faster (possibly connected to stars formed in-situ/in large building blocks) and a more metal-poor one that rotates less fast or not at all (possibly connected to stars that were accreted/formed in smaller building blocks). This is supported by a preferred two-component fit to the velocity distribution for stars with -2.0 < [Fe/H] < -1.5, with parameters similar to the two halo components recently found by Zhang et al. (2023).

(v) The high velocity dispersion of metal-poor inner Galaxy stars is mostly driven by ‘halo interlopers’, and previous PIGS results of a strongly rising velocity dispersion with decreasing metallicity (Arentsen et al. 2020a) can be explained by this (see Fig. 11, consistent with previous findings by Lucey et al. 2021 and Kunder et al. 2020). After removing stars with apocentres larger than 5 kpc, the velocity dispersion is still rising with decreasing metallicity, but only gently: from ~80 km s⁻¹ for [Fe/H] = -1 to ~95 km s⁻¹ for [Fe/H] < -2.0. This is likely due to the VMP inner Galaxy stars having a slightly more extended distribution compared to less metal-poor stars.

(vi) We investigated the orbital properties of PIGS CEMP stars in the inner Galaxy (Arentsen et al. 2021), and found no significant difference between the apocentre distributions of carbon-rich and carbon-normal VMP stars (Fig. 12). More work is necessary to further investigate whether the occurrence of CEMP stars is different between ‘true inner Galaxy stars’ and ‘halo interlopers’.

The past years have seen significant progress in the study of the metal-poor inner Galaxy, but there are still many open questions about the ancient stellar population(s) in this region. The next years will see many new results from several large spectroscopic surveys targeting the Galactic bulge (e.g. SDSS-V, MOONS, 4MOST), with the 4MOST surveys 4MIDABLE-LR and -HR (Bensby et al. 2019; Chiappini et al. 2019) being particularly promising for the investigation of the metal-poor inner Milky Way. Large, homogeneous samples of metal-poor stars with chemical abundances will allow us to probe the chemical evolution in the earliest phase of our Galaxy.

ACKNOWLEDGEMENTS

We thank the reviewer for their helpful suggestions. AAA thanks
all members of the Pristine collaboration and the Cambridge
Streams group for valuable discussions. We thank Sarah Martell
and Jeffrey Simpson for observing for the PIGS/AAT program.
AAA acknowledges support from the Herchel Smith Fellowship at
the University of Cambridge and a Fitzwilliam College research
fellowship supported by the Isaac Newton Trust. AAA, GM, NFM,
and ZY gratefully acknowledge funding from the European Research
Council (ERC) under the European Unions Horizon 2020 research
and innovation programme (grant agreement No. 834148). GM,
NFM, and ZY gratefully acknowledge support from the French National Research Agency (ANR) funded projects "MWdisc" (grant N211483) and "Pristine" (ANR-18-CE31-0017). ES acknowledges funding through VIDI grant "Pushing Galactic Archaeology to its limits" (with project number VI.Vidi.193.093) which is funded by the Dutch Research Council (NWO). This research has been partially funded from a Spinoza award by NWO (SPI 78-411). DA acknowledges financial support from the Spanish Ministry of Science and Innovation (MICINN) under the 2021 Ramón y Cajal program MICINN RYC2021-032609. GCM thanks the Leverhulme Trust and the Isaac Newton Trust for their support on his research. HZ thanks the Science and Technology Facilities Council (STFC) for a PhD studentship. WHO acknowledges financial support from the Carl Zeiss Stiftung and the Paulette Isabel Jones PhD Completion Scholarship at the University of Sydney.

This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project 540 (The Early Milky Way).

We thank the Australian Astronomical Observatory, which have made the PIGS spectroscopic follow-up observations used in this work possible. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. Based on data obtained at Siding Spring Observatory (via programs S/2017B/01, A/2018A/01, OPTICON 2018B/029, and OPTICON 2019A/045, PI: A Arentsen and A/2020A/11, PI: DB Zucker).

The spectroscopic follow-up used in this work was based on selection from observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.

This research made extensive use of the MATPLOTLIB (Hunter 2007), PANDAS (McKinney 2010), and ASTROPY (Astropy Collaboration 2013; Price-Whelan et al. 2018) PYTHON packages, and of TOPCAT (Taylor 2005). We thank Sergey Koposov for sharing his column-density normalized 2D histogram PYTHON code.

**Author Contributions Statement:** AAA is the PI of PIGS and initiated and designed the study, performed the analyses, made all the figures and led the writing of the manuscript. GM derived the orbital properties for the sample. AQ derived the starhorse distances for the sample. Both wrote the respective method sections in the manuscript. ES, NFM and CC contributed through numerous discussions and assisted in the development of the project and the manuscript. DA derived the PIGS/AAT spectroscopic parameters with FERRE. AAA, NFM, SB, GFL, WHO, ZW and DBZ performed the observations at the AAT. DBZ also assisted in preparing for the AAT observations. All authors provided feedback and helped shape the manuscript.

## DATA AVAILABILITY

We release the PIGS spectroscopic parameters (methodology described in Arentsen et al. 2020b) with this paper, as well as the distances and orbital properties derived in this work (summarized by the median and 16th + 84th percentiles, as well as the complete samples). They are available at the CDS. The STARHORSE Gaussian Mixture model for the distance solutions are available on request to A ArdBern-Arentsen.

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This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project 540 (The Early Milky Way).

We thank the Australian Astronomical Observatory, which have made the PIGS spectroscopic follow-up observations used in this work possible. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. Based on data obtained at Siding Spring Observatory (via programs S/2017B/01, A/2018A/01, OPTICON 2018B/029, and OPTICON 2019A/045, PI: A Arentsen and A/2020A/11, PI: DB Zucker).

The spectroscopic follow-up used in this work was based on selection from observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

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This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.

This research made extensive use of the MATPLOTLIB (Hunter 2007), PANDAS (McKinney 2010), and ASTROPY (Astropy Collaboration 2013; Price-Whelan et al. 2018) PYTHON packages, and of TOPCAT (Taylor 2005). We thank Sergey Koposov for sharing his column-density normalized 2D histogram PYTHON code.

**Author Contributions Statement:** AAA is the PI of PIGS and initiated and designed the study, performed the analyses, made all the figures and led the writing of the manuscript. GM derived the orbital properties for the sample. AQ derived the starhorse distances for the sample. Both wrote the respective method sections in the manuscript. ES, NFM and CC contributed through numerous discussions and assisted in the development of the project and the manuscript. DA derived the PIGS/AAT spectroscopic parameters with FERRE. AAA, NFM, SB, GFL, WHO, ZW and DBZ performed the observations at the AAT. DBZ also assisted in preparing for the AAT observations. All authors provided feedback and helped shape the manuscript.

## DATA AVAILABILITY

We release the PIGS spectroscopic parameters (methodology described in Arentsen et al. 2020b) with this paper, as well as the distances and orbital properties derived in this work (summarized by the median and 16th + 84th percentiles, as well as the complete samples). They are available at the CDS. The STARHORSE Gaussian Mixture model for the distance solutions are available on request to A ArdBern-Arentsen.

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## SUPPORTING INFORMATION

Supplementary data are available at MNRAS online.

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

This paper has been typeset from a TeX/LA TeX file prepared by the author.

<footnote n="1">¹Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="2">²Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France</footnote>

<footnote n="3">³Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="4">⁴Universidad de La Laguna, Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="5">⁵Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany</footnote>

<footnote n="6">⁶Kapteyn Astronomical Institute, University of Groningen, Postbus 800, NL-9700 AV, Groningen, the Netherlands</footnote>

<footnote n="7">⁷Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany</footnote>

<footnote n="8">⁸Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada</footnote>

<footnote n="9">⁹Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Blvd de l'Observatoire, F-06304 Nice, France</footnote>

<footnote n="10">¹⁰Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada</footnote>

<footnote n="11">¹¹Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile</footnote>

<footnote n="12">¹²Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia</footnote>

<footnote n="13">¹³ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia</footnote>

<footnote n="14">¹⁴Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, NSW 2006, Australia</footnote>

<footnote n="15">¹⁵Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany</footnote>

<footnote n="16">¹⁶Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany</footnote>

<footnote n="17">¹⁷School of Astronomy and Space Science, University of Science and Technology of China, 230026 Hefei, China</footnote>

<footnote n="18">¹⁸School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, Australia</footnote>

<footnote n="19">¹⁹Macquarie University Astrophysics and Space Technologies Research Centre, Sydney, NSW 2109, Australiac</footnote>
Wolf C. et al., 2018, PASA, 35, e010

Wright E. L. et al., 2010, AJ, 140, 1868

Wylie S. M., Gerhard O. E., Ness M. K., Clarke J. P., Freeman K. C., Bland-Hawthorn J., 2021, A&A, 653, A143

Yang C., Zhu L., Tahmasebzadeh B., Xue X.-X., Liu C., 2022, AJ, 164, 241

Youakim K. et al., 2020, MNRAS, 492, 4986

Yuan Z. et al., 2023, preprint (arXiv:2311.08464)

Zhang H., Ardern-Arentsen A., Belokurov V., 2023, preprint (arXiv:2311.09294)

Zoccali M. et al., 2017, A&A, 599, A12

## SUPPORTING INFORMATION

Supplementary data are available at MNRAS online.

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

<footnote n="1">¹Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="2">²Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France</footnote>

<footnote n="3">³Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="4">⁴Universidad de La Laguna, Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="5">⁵Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany</footnote>

<footnote n="6">⁶Kapteyn Astronomical Institute, University of Groningen, Postbus 800, NL-9700 AV, Groningen, the Netherlands</footnote>

<footnote n="7">⁷Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany</footnote>

<footnote n="8">⁸Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada</footnote>

<footnote n="9">⁹Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Blvd de l'Observatoire, F-06304 Nice, France</footnote>

<footnote n="10">¹⁰Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8W 3P2, Canada</footnote>

<footnote n="11">¹¹Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile</footnote>

<footnote n="12">¹²Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia</footnote>

<footnote n="13">¹³ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia</footnote>

<footnote n="14">¹⁴Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, NSW 2006, Australia</footnote>

<footnote n="15">¹⁵Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany</footnote>

<footnote n="16">¹⁶Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany</footnote>

<footnote n="17">¹⁷School of Astronomy and Space Science, University of Science and Technology of China, 230026 Hefei, China</footnote>

<footnote n="18">¹⁸School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, Australia</footnote>

<footnote n="19">¹⁹Macquarie University Astrophysics and Space Technologies Research Centre, Sydney, NSW 2109, Australiac</footnote>

<footnote>This paper has been typeset from a TeX/LA TeX file prepared by the author.</footnote>