adsabs/FOCAL huggingface dataset:
Figures 4 and 5 display two enlarged regions from Figure 3, Region 1 being quiet Sun and Region 2 being plage. We have plotted a variety of statistics in these plots that measure the components of the profiles. The emission at h1v generally originates in the upper photosphere. Based on inspection of this and additional higher spatial resolution IRIS data sets, we determine that the structures are a mix of reverse granulation and grains. At h2v, much of the small granular structure disappears and a diffuse halo of bright structures overlay and surround magnetic concentrations. The brightest plage is almost twice as bright as the brightest network, and the maximum h2v emission occurs near the center of the magnetic flux concentration. Single-peaked profiles are relativity common (∼10%) in plage, as depicted by the contiguous red regions in Figure 4. Figure 6(a) shows an example of a single-peaked plage profile. While our algorithm separates between single-peaked and double-peaked profiles to aid in statistical analysis, there is in fact a smooth continuum of weakly separated, weakly reversed profiles that bridge those two profile categories (Figure 6(c) shows an example). In the semi-empirical models of the solar atmosphere, the Mg ii h source function above the temperature minimum has a local maximum at an altitude of 1.2 Mm (see Figure 7 in Leenaarts et al. 2013a), while the line core forms near a height of 2 Mm. To produce a single-peaked profile the atmosphere must be structured such that source function does not vary significantly between 





 at h2 and h3. At high densities, the source function closely adheres to the Planck function, which rises through the chromosphere. These profiles are consistent with the model that hot high-density loops are rooted in plage. Most sunspot profiles are single peaked, and an example is displayed in Figure 6(e). Single-peaked umbral profiles were previously reported in Lites & Skumanich (1982) and Morrill et al. (2001). Sunspot profiles are half as bright at h2 and 20% narrower at h1 than plage. The far wing intensity (beyond h1) is lower in sunspots than plage or internetwork. Single-peaked profiles also occur in internetwork regions, albeit at much lower frequency (∼1%). Figure 6(b) shows a typical internetwork single-peaked profile. Although this profile contains a single maximum, it still requires a substantial contribution from the negative Gaussian amplitude (h in Equation (1)) to achieve a low 





 Figures 6(d) and (f) show profiles that occur at low frequencies in the internetwork (





). These profiles represent some of the extreme variations from the mean profile shape that we observe in this large data set. The profile of Figure 6(d) could be produced by extreme upflows in the upper chromosphere that completely mask the photons in the nominal position of the h2v peak and wing. Figure 6(f) illustrates Mg ii h in absorption, which requires that the source function monotonically decrease from the photosphere precluding the mid-chromospheric temperature rise.
Citation: Leenaarts et al. 2013a
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For case 1, we find that most of the stellar values of the MW halo ([Fe/H]  −1) reside between the evolutionary tracks explored here and also in the coloured region. Note that the carbon stars are excluded in Fig. 5, since they might have been contaminated by the binary mass transfer from former asymptotic-giant-branch stars. The smaller SFE for a less-massive galaxy leads to an increase of [Eu/Fe] at a lower [Fe/H]. It is notable that our model predicts that almost all highly r-process-enhanced stars ([Eu/Fe] > 1 at [Fe/H] ∼ −3) originate from UFD-sized (M*  105) building blocks [but see a caution in Section 4 for the applicability of the mass–metallicity relation of Kirby et al. (2013) to UFD-sized systems], as also suggested in Ishimaru et al. (2015) and Ojima et al. (2018). For instance, our model suggests that J1521−3538 (star, the highest measured Eu/Fe, Cain et al. 2020) and CS 31082−001 (diamond, Hill et al. 2002) were born in building-block galaxies of M* ∼ 103 and 5 × 103 (in M⊙), respectively. This is a consequence of the fact that a less-massive galaxy contains a smaller amount of gas to be mixed with Eu from an NSM. In fact, the highly eccentric orbit of the former indicates that J1521−3538 originates from a building-block galaxy subsequently accreted by the MW (Cain et al. 2020). In contrast, the r-process-deficient stars may have been born in the most-massive building-block galaxies of M* ∼ 108 M⊙, like HD 4306 (circle, the lowest measured Eu/Fe, Ishimaru et al. 2004) and HD 122563 (triangle, Honda et al. 2006). Note that our model predicts the presence of even further r-process-deficient stars with [Eu/Fe]  −1 at [Fe/H] ≲ −3. The absence of such stars with measured Eu is likely due to the current detection limit for Eu (Ishimaru et al. 2004). Overall, the result for case 1 (a mean delay of 0.1 Gyr in the lognormal term for NSMs) appears in qualitative agreement with those in Ishimaru et al. (2015) and Ojima et al. (2018) with a fixed delay of 0.1 Gyr (see Appendix A for additional tests).
Citation: Hill et al. 2002
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Citation: Ojima et al. (2018)
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Citation: Ojima et al. (2018)
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Citation: Honda et al. 2006
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As to the chemical enrichment history of individual galaxies, we have exploited the standard code che-evo incorporated into GRASIL (Silva et al. 1998, 2011; Bressan et al. 2002; Panuzzo et al. 2003; Vega et al. 2005). For spheroidal galaxies, it reproduces the observed local relationship between stellar metallicity and stellar mass and its weak evolution with redshift (see Arrigoni et al. 2010; Spolaor et al. 2010; Gallazzi et al. 2014). For disk galaxies at z ≲ 2, it reproduces the observed relationship between gas metallicity and stellar mass, including its appreciable redshift dependence (see Andrews & Martini 2013; Zahid et al. 2014; de la Rosa et al. 2016; Onodera et al. 2016). In both cases, within an individual galaxy the metallicity behavior is closely approximated by
4





i.e., it increases from Z = 0 almost linearly with the galactic age, and then after a time τ = Δτψ it saturates to the value Zsat. The dependence of Zsat and Δ on SFR/stellar mass can be described with an expression inspired by analytic chemical evolution models (see Cai et al. 2013; Feldmann 2015); this yields
5





here s ≈ 3 is the ratio between the dynamical timescale of the infalling gas and the star formation timescale, 





 is the recycling stellar mass fraction, 





 is the metal yield (assuming the Romano et al. 2010 stellar yields), and out is the mass-loading factor of galactic outflows from stellar winds and SN explosions. In the above equation we have provided a fit for out as a function of the final stellar mass M⋆ from the results of the che-evo code; a similar expression concurrently describes the time-averaged outcome from hydrodynamical simulations of stellar feedback (e.g., Hopkins et al. 2012). As a result, typical values 





 are obtained for galaxies with final stellar masses in the range M⋆ ∼ 109–1011 M⊙, respectively (see, e.g., Chruslinska et al. 2019); the related quantity Δ ∼ 0.1–0.3 specifies how quickly the metallicity saturates to such values as a consequence of the interplay between cooling, dilution, and feedback processes. Note that several chemical evolution codes present in the literature, reproducing comparably well observations on the chemical abundances in galaxies of different stellar masses, also share a similar age-dependent metallicity behavior.
Citation: Zahid et al. 2014
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Citation: Cai et al. 2013
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Citation: Vega et al. 2005
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Citation: Hopkins et al. 2012
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Standard data reduction was applied to the data taken with HIPO (red and blue) and FPI+. This includes bias and dark subtraction, and in the case of FPI+ also flat field correction. For FLITECAM we did not acquire bias frames, as bias contributions are generally very low for this type of NIR detector array. Furthermore, it is complicated and time consuming to obtain a reliable flat field on such a narrow band filter as the 1.9 μm Paschen-α continuum filter. Since it was not possible to take long enough exposures during this campaign, we used K-band flat fields taken on the same flight before our observation run, which, however, did not improve the photometric precision significantly. Similarly, dark subtraction did not show any improvement in photometric precision. Additionally, we corrected for the sky background using dithered images taken during the observation run. Stars were detected with the Source Extractor by Bertin & Arnouts (1996). Aperture photometry was applied using Image Reduction and Analysis Facility’s (IRAF Tody 1993) DAOPHOT (Stetson 1987) using circular apertures. The optimal aperture radius with the lowest noise level was found to be six pixels for FLITECAM and 12 pixel for HIPO and FPI+. As part of the DAOPHOT routine, an annulus around the target was used to estimate the sky background in each exposure. Next to GJ 1214, we extracted the light curves of two additional bright stars within our field of view (see Table 1 and Fig. 1). To identify stars in the images we calculated a rough astrometric solution for each image using data provided by Astrometry.net (Barron et al. 2008). In the FLIPO setup SOFIA does not provide an image rotator to compensate field rotation during long integrations. This introduces a rotation of the images over time. Due to SOFIA’s unique setup, the telescope must periodically undergo so-called “line-of-sight (LOS) rewinds”. The required frequency of LOS rewinds depends on the rate of field rotation experienced by the target, which is a complex function of the position of the target in the sky relative to that of the aircraft heading. These need to be carefully timed with regard to the transit observation, to not interfere with, for example, ingress or egress. While we kept the target star, GJ 1214, in boresight, the comparison stars moved over the CCD due to this field rotation. This is one of the main factors introducing systematic noise and limiting the photometric precision of the instrument and is another reason why reliable flat fields are crucial for this kind of time series observation. After comparing the light curves, the brighter star was selected as comparison star to correct for first-order systematic effects present in all light curves.
Citation: Bertin & Arnouts (1996)
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Citation: Barron et al. 2008
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Dust grains are modelled using Lagrangian tracer particles, where each simulated particle represents a collection of dust grains with similar properties and motions. Trajectories of the particles are integrated using the fully implicit method of Bai & Stone (2010), which we have incorporated into the VL integrator in Athena. In a Cartesian coordinate system, Athena solves an equation of motion for each particle given by

(12)

\begin{equation}
\frac{{\rm d} {{\boldsymbol v}_i}}{{\rm d}t} = -\frac{{{\boldsymbol v}_i} - {\boldsymbol u}}{t_{\rm stop}},
\end{equation}

with ${{\boldsymbol v}_i}$ the velocity vector of particle i, ${\boldsymbol u}$ the local gas velocity vector, and tstop the particle stopping time due to gas drag. Neglecting grain charges and assuming only pure hydrogen gas, the (collisional) drag law is given by (Draine & Salpeter 1979)

(13)

\begin{equation}
\frac{{\rm d} {\boldsymbol v}_i}{{\rm d}t} \approx -\frac{2 \pi a^2 n k_{\rm B} T G_0(s)}{(4/3)\pi \rho _{\rm d} a^3},
\end{equation}

with

(14)

\begin{equation}
G_0(s) \approx \frac{8 s}{3 \sqrt{\pi }} \left(1 + \frac{9\pi }{64} s^2\right)^{1/2}
\end{equation}

and

(15)

\begin{equation}
s \equiv \left(\frac{m_{\rm H} {\boldsymbol v}_{\rm rel}^2}{2 k_{\rm B} T}\right)^{1/2},
\end{equation}

where a is the dust grain radius, kB is the Boltzmann constant, T is the temperature of the gas, n is the gas number density, ρd is the internal density of the dust (which we treat as constant at ρd = 3.0 g cm−3), mH is the mass of hydrogen, and ${\boldsymbol v}_{\rm rel} \equiv {{\boldsymbol v}_i} - {\boldsymbol u}$ is the relative velocity difference between the dust and gas. The stopping distance is evaluated as

(16)

\begin{equation}
t_{\rm stop} = \frac{\sqrt{\pi }}{2\sqrt{2}} \frac{a \rho _{\rm d}}{n \sqrt{m_{\rm H} k_{\rm B} T}} \left(1 + \frac{9\pi m_{\rm H}}{128 k_{\rm B} T} {\boldsymbol v}^2_{\rm rel}\right)^{-1/2}.
\end{equation}

The gas properties (n, T, ${\boldsymbol u}$) at each particle's location are calculated from nearby grid points using a triangular-shaped cloud (TSC) interpolation scheme (Hockney & Eastwood 1988). There is no momentum feedback from the particles on the gas.
Citation: Draine & Salpeter 1979
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We scale the maps of NGC 4710 to a system that can be directly compared with the Milky Way bulge and compare the resulting maps with those constructed by Zoccali et al. (2014) based on interpolation of the GIBS survey rotational curves. In this comparison we found that the rotation map of NGC 4710 and the rotation map of the Milky Way bulge show a remarkable similarity. Although the velocity dispersion map of the bulge of NGC 4710 appears noisier than that of the Milky Way bulge, owing to the way the maps are constructed, we see that the velocity dispersion map of the Milky Way bulge is more vertically elongated in the centre than in the velocity dispersion map of NGC 4710. We see a similar change in the velocity dispersion maps constructed using different bar orientation angles in the simulation from Cole et al. (2014). This is also in good agreement with the features identified in the LOS kinematic maps of BP bulges modelled by Iannuzzi & Athanassoula (2015). We thus suggest that central increase in the velocity dispersion profile of the Milky Way bulge can be partially the consequence of the Milky Way bar and BP bulge viewing angle of 27° with respect to the Sun-Galactic centre LOS. However, a comparison of the rotation and velocity dispersion curves at different heights from the plane, used by Zoccali et al. (2014) to obtain the rotation maps, are very similar to those of the bulge of NGC 4710. These 1D rotation profiles are also in good agreement with the simulation of a galaxy hosting a pure BP bulge. The differences identified between the inner regions of the σ maps of the Milky Way and NGC 4710 are not seen in the 1D curves. This suggests that the vertical elongation observed in the central region of the σ map of the Milky Way bulge could be an artefact from the plane interpolation method to account for the σ peak observed at b = −2. Instead, the σ peak in the Milky Way bulge would be limited to Galactic latitudes | b |   2 (~0.28 kpc).
Citation: Iannuzzi & Athanassoula (2015)
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Citation: Zoccali et al. (2014)
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Citation: Zoccali et al. (2014)
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The third and higher harmonic plasma emissions, at the frequency of multiples of the local plasma frequency, were also observed in SRBs (Takakura & Yousef 1974; Cairns 1986; Reiner & MacDowall 2019). Various schemes have been proposed to explain the formation of higher harmonic plasma emission. Zlotnik (1978) put forward that the third harmonic emission H
3 is generated by the coalescence process of the Langmuir waves and the harmonic emission H, i.e., L + H → H
3. Cairns (1988) generalized this idea to explain higher harmonics of transverse wave emissions, claiming that nth harmonic waves H

n
 could be generated by a coalescence of beam-generated Langmuir waves L and adjacent transverse waves H

n−1, i.e., L + H

n−1 → H

n
. An alternative way for the generation of harmonic transverse waves would be an interaction of Langmuir waves L and harmonic electrostatic waves L

n−1, i.e., L + L

n−1 → H

n
 (Gaelzer et al. 2003; Yi et al. 2007). Ziebell et al. (2015) numerically solved the weak-turbulence equations and theoretically demonstrate the process of multiple-harmonic plasma emission, including the nonlinear conversion from Langmuir turbulence to electromagnetic radiation. Rhee et al. (2009), Ziebell et al. (2015), Yao et al. (2021) investigated both schemes of multiple-harmonic plasma emissions by means of Particle-In-Cell (PIC) code simulations. They found that Cairns (1988)'s scheme can qualitatively explain the third and fourth harmonic emissions while Yi et al. (2007)'s theory is appropriate to explain the simultaneously generated harmonics of electrostatic waves. PIC code simulations of electron beams demonstrated that fundamental transverse waves at the plasma frequency can be generated because of the wave–wave interaction processes involving ion-acoustic waves (Thurgood & Tsiklauri 2015; Henri et al. 2019). The PIC simulations of ring-beam EVDFs by Zhou et al. (2020) also provided evidence for the generation of the second and third harmonic emission via wave–wave interactions. Annenkov et al. (2019) used PIC code simulations to describe localized beams and found that the emissions at harmonics of the plasma frequency can be generated due to an antenna mechanism.
Citation: Gaelzer et al. 2003
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Measurements of the scale length (h

R
) and scale height (h

Z
) are important to trace the structure, size, mass distribution, and radial luminosity profile of the Galactic disk components (e.g., Dehnen & Binney 1998). In order to calculate h

R
 and h

Z
 of our Atari disk sample, we solve the fundamental collisionless Boltzmann equation of axisymmetric systems, which is expressed as follows (see Equation (4.12) in Binney & Tremaine 2008):
10



∂f∂t+vR∂f∂R+vϕR2∂f∂ϕ+vz∂f∂z−∂Φ∂R−vϕ2R3∂f∂vR−∂Φ∂ϕ∂f∂vϕ−∂Φ∂z∂f∂vz=0,

where f is the number of objects in a small volume and Φ is the gravitational potential. It is then convenient to derive the Jeans equation from the Boltzmann equation in the radial and Z-component directions as follows (see Equation (9) in Gilmore et al. 1989):
11



ρKR=1R∂(RρσVR2)∂R+∂(ρσVR,Z2)∂Z−ρσVϕ2R−ρRVϕ¯2



12



ρKZ=∂(ρσVZ2)∂Z+1R∂(RρσVR,Z2)∂R,

where ρ(R, Z) is the space density of the stars in the thick disk and K

R
 = 



∂ϕ∂R

 and K

Z
 = 



∂ϕ∂Z

 are the derivatives of the potential. Assuming an exponential density profile, the radial Jeans equation can be rewritten as follows (Li et al. 2018):
13



σVϕ2σVR2−2+2RhR−Vc2−Vϕ¯2σVR2+σVZ2σVR2=0,

where h

R
 is the scale length. By substituting our calculated velocity dispersions from the Atari disk sample, within ≈3 kpc of the Sun in the cylindrical R coordinate and ≈2 kpc above or below the Galactic plane (6347 stars), into Equation (13), we obtain a radial scale length of h

R
 = 2.48 kpc. Calculating the scale length using different metallicity bins shows a small increase from 2.38 kpc among the higher-metallicity stars up to 2.91 kpc for the low-metallicity stars. The results are detailed in Table 3. In general, these results point to the Atari disk being comparable in size in the radial direction to the thick and thin disk. For reference, the scale length of the canonical thick disk has been measured as 2.0 kpc (Bensby et al. 2011), 2.2 kpc (Carollo et al. 2010), and 2.31 kpc (Sanders & Binney 2015), although larger values have also been reported previously (Chiba & Beers 2000; de Jong et al. 2010). Thin disk values refer to an overall similar spatial distribution, although it is likely somewhat more extended (h

R
 > 3.0 kpc;  e.g., Bensby et al. 2011; Sanders & Binney 2015). See Table 2 for further details.
Citation: Li et al. 2018
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For the cosmic variance term, any two samples are correlated so the off-diagonal
            element of Gij
              (i ≠
              j) is non-zero. But for the measurement noise term,
            small scale, and intrinsic dispersion, the two samples are not correlated, so they only
            contribute to the diagonal term in the Gij matrix. The first
            term is the real space velocity correlation function, which is related to the matter
            power spectrum in Fourier space (Watkins et al.
              2009; Ma & Scott 2013),
              (10)\begin{eqnarray} \bigl\langle \bigl(\hat{{\vec r}}_{i} \cdot {\vec v}({\vec r}_{i})\bigr)  \bigl(\hat{{\vec r}}_{j} \cdot {\vec v}({\vec r}_{j})\bigr) \bigr\rangle  =\frac{H^{2}_{0}f^{2}(z=0)}{2 \pi^{2}}\int \mathrm{\ } \der k  \mathrm{\ } P(k) \mathrm{\ } F_{ij}(k), \label{vel_rnrm1} \end{eqnarray}

















r


ˆ


i

·


v


(



r



i


)













r


ˆ


j

·


v


(



r



j


)










=






H

0


2



f

2


(
z
=
0
)


2

π

2








∫




d
k



P
(
k
)




F

ij


(
k
)
,










where P(k) is
            the matter power spectrum that we output from public code camb (Lewis et al. 2000). The f(z) =
                dlnD/ dlna is
            the growth rate function that characterizes how fast the structures grow at different
            epochs of the Universe. Since the Cosmicflows-2  samples peak at the redshift
              z ≃
              0.0167 (lower panel of Fig. 2),
            we use the zero-redshift growth function \hbox{$f(z=0)=\Omega_{\rm m}^{0.55}$}
f
(
z
=
0
)
=

Ω
m
0.55

 (Watkins
              et al. 2009; Ma & Scott 2013) in Eq.
              (10). In the future, if the survey
            probes deeper region of the space, one should use the corresponding growth function
              f(z) in Eq. (10), so that the joint constraints on
              f(z)σ8
            can be obtained, which constitutes a sensitive test of modify gravity models (Hudson & Turnbull 2012; Planck Collaboration XIII 2015).
Citation: Ma & Scott 2013
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Citation: Ma & Scott 2013
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