# A and F stars as probes of outer Galactic disc kinematics

A. Harris,1 J. E. Drew1, H. J. Farnhill1, M. Monguió1, M. Gebran2, N. J.
Wright3, J. J. Drake4, S. E. Sale5

1School of Physics, Astronomy and Mathematics, University of Hertfordshire,
College Lane, Hatfield AL10 9AB, UK
2Department of Physics and Astronomy, Notre Dame University - Louaize, PO Box
72, Zouk Mikael, Lebanon
3Astrophysics Group, Keele University, Keele ST5 5BG, UK
4Harvard Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
5Astrophysics Group, School of Physics, University of Exeter, Stocker Road,
Exeter EX4 4QL, UK<EMAIL_ADDRESS>
(Accepted 2017 December 19. Received 2017 December 15; in original form 2017
October 27)

###### Abstract

Previous studies of the rotation law in the outer Galactic disc have mainly
used gas tracers or clump giants. Here, we explore A and F stars as
alternatives: these provide a much denser sampling in the outer disc than gas
tracers and have experienced significantly less velocity scattering than older
clump giants. This first investigation confirms the suitability of A stars in
this role. Our work is based on spectroscopy of $\sim$ 1300 photometrically-
selected stars in the red calcium-triplet region, chosen to mitigate against
the effects of interstellar extinction. The stars are located in two low
Galactic latitude sightlines, at longitudes $\ell=118^{\circ}$, sampling
strong Galactic rotation shear, and $\ell=178^{\circ}$, near the Anticentre.
With the use of Markov Chain Monte Carlo parameter fitting, stellar parameters
and radial velocities are measured, and distances computed. The obtained trend
of radial velocity with distance is inconsistent with existing flat or slowly
rising rotation laws from gas tracers (Brand & Blitz, 1993; Reid et al.,
2014). Instead, our results fit in with those obtained by Huang et al. (2016)
from disc clump giants that favoured rising circular speeds. An alternative
interpretation in terms of spiral arm perturbation is not straight forward. We
assess the role that undetected binaries in the sample and distance error may
have in introducing bias, and show that the former is a minor factor. The
random errors in our trend of circular velocity are within $\pm 5$ km s-1.

###### keywords:

Galaxy: disc – Galaxy: kinematics and dynamics – techniques: radial velocities
– methods: observational

††pagerange: A and F stars as probes of outer Galactic disc kinematics–A and F
stars as probes of outer Galactic disc kinematics††pubyear: 2017

##  1 Introduction

The kinematics of stars outside the Solar Circle is relatively poorly known.
In addition to setting a constraint on the Galactic potential, secure
knowledge of the motions of stars at all radii and longitudes in the Galactic
disc facilitates studies of non-axisymmetric motion, such as streaming motions
due to spiral density waves or transient winding arms. Understanding the
kinematics of the disc in full, including the rotation law, will help us map
out its structure, and set constraints on its formation and evolution.

Within the disc of the Milky Way, the rotational motion is the dominant
component. The IAU recommended standard of the circular velocity at the Sun is
$220$ km s-1 (Kerr & Lynden-Bell, 1986). This result is based mainly on the
analysis of tracers of the gaseous interstellar medium (ISM), including CO, H
i 21 cm and H ii radio recombination line observations. In recent decades the
work of Brand & Blitz (1993) has been particularly influential. How the
rotation varies outward from the Solar Circle is more challenging to measure
than within because of the greater reliance on the uncertain distances to the
ISM tracers.

It has been common practice to associate CO and H ii radio recombination lines
with star forming regions whose distances are specified by spectroscopic
parallax. More recent studies have made use of masers in star forming regions
with much better defined distances, and these have begun to favour a higher
rotation speed. Honma et al. (2012) used a sample of 52 masers associated with
pre-main sequence stars to find the rotational velocity of the local standard
of rest (LSR) to be between $223$ km s-1 and $248$ km s-1, with a flat
rotation curve. Reid et al. (2014) took a similar approach, using an expanded
sample of over 100 masers to refine the circular rotation speed at the Sun to
$240\pm 8$ km s-1. Bobylev, Bajkova & Shirokova (2016) support a raised
circular speed ($236\pm 6$ km s-1) and a nearly flat rotation curve, based on
a large sample of open clusters.

A feature of the studies to date is a drop in sensitivity with increasing
Galactocentric radius, RG, as the number or quality of available measurements
declines. In the case of maser measurements, just a handful are available
beyond R${}_{G}\sim 12$ kpc. Similarly, the number of star clusters catalogued
at R${}_{G}\geq 10$ kpc is limited to a few tens. This is an unavoidable
consequence of the inherent low spatial frequency of these object classes
outside the Solar Circle. Hence, Galactic rotation of the outer disc remains
highly uncertain.

The frequency of stars at increasingly large Galactocentric radii becomes much
higher than either masers or H ii regions and work has been undertaken
exploiting helium-burning clump giants that can serve as a form of standard
candle (Castellani, Chieffi & Straniero, 1992). Examples of this, emphasising
measurement of the outer disc, are the works by López-Corredoira (2014), using
proper motions from PPMXL111PPMXL (Roeser, Demleitner & Schilbach, 2010) is a
catalog of positions and proper motions referred to the ICRS (International
Celestial Reference System) and by Huang et al. (2016) using LSS-GAC222LAMOST
Spectroscopic Survey of the Galactic Anticentre (Xiang et al., 2017). LAMOST
is the Large Sky Area Multi-Object Fibre Spectroscopic Telescope. and APOGEE
333The Apache Point Observatory Galactic Evolution Experiment (DR12 release,
see Alam et al., 2015) radial velocities. As older objects with ages exceeding
1 Gyr, clump giants will be subject to significant kinematic scatter,
including asymmetric drift.

In this study we examine another class of stellar tracer that can also provide
a much denser sampling of the outer disc than is available from ISM tracers.
We turn to A/F type stars. These stars offer the following advantages: they
are intrinsically relatively luminous, with absolute magnitudes in the $i$
band of $\sim$0 to 3; as younger objects ($\sim 100$ Myr), they have
experienced significantly less scattering within the Galactic disc (Dehnen &
Binney, 1998); and, as we shall show, the A stars especially are efficiently
selected from photometric H$\alpha$ surveys. We are able to detect them at
useful densities out to R${}_{G}\sim$14 kpc. Furthermore, the numbers of A
type stars most likely exceed the numbers of clump giants in the outer disc,
given that the outer disc is younger than the inner disc (Bergemann et al.,
2014). Our own investigations of photometry from the INT 444Isaac Newton
Telescope Photometric H$\alpha$ Survey of the Northern Galactic Plane (IPHAS,
Drew et al., 2005), used here for target selection, supports this expectation.

We explore radial velocity (RV) measurements of near-main sequence A/F stars
as kinematic tracers in two outer-disc pencil beams, with a view to future
fuller exploitation via spectroscopy on forthcoming massively multiplexed
wide-field spectrographs (e.g WEAVE555WHT (William Herschel Telescope)
Enhanced Area Velocity Explorer (Dalton et al., 2016), in construction for the
William Herschel Telescope). To greatly reduce the impact of extinction on the
reach of our samples, our observations target the calcium triplet (CaT)
region, in the far red part of the spectrum. Now is a good time to embark on
this investigation in view of the imminent release of Gaia astrometry, which
will include complementary proper motions of disc stars and support improved
distance estimates.

We present a method of RV measurement in the CaT part of the spectrum and
apply it to a dataset of $\sim 1300$ spectra of A and F type field stars, in
order to confirm their viability as tracers of Galactic disc kinematics and to
pave the way for more comprehensive use of them in future. This first dataset
is made up of stars in two pencil beams at Galactic longitudes
$\ell=118^{\circ}$ and $\ell=178^{\circ}$. Our data were obtained at MMT using
the multi-object spectrograph HectoSpec. Section 2 describes the observations
obtained and how they were prepared for analysis. In section 3 we detail the
method used to obtain the RVs and stellar parameters and determine distances.
The derived parameter distributions and error budgets are presented. The
results obtained close to the anticentre at $\ell=178^{\circ}$, our control
direction, demonstrate the validity of the analysis, while the results at
$\ell=118^{\circ}$ indicate a strong departure from a flat or slowly-rising
Galactic rotation law (section 4). In section 5, we review possible bias, make
comparisons with earlier results and consider whether structure due to spiral
arm perturbations may have been detected. Our results turn out to be
consistent with the rising section of the rotation law presented by Huang et
al. (2016) based on disc clump giants over the Galactocentric radius range 11
to 15 kpc. A summary of our conclusions is given in section 6.

##  2 Observations and data

###  2.1 Sightline and selection of targets for observation

We chose to study two pencil-beam sightlines located at Galactic co-ordinates
$\ell=178^{\circ}$, $b=1^{\circ}$ and $\ell=118^{\circ}$, $b=2^{\circ}$. These
were chosen for the following reasons:

  * •

In principle, RVs measured at $l=118^{\circ}$ should sample strong shear in
Galactic rotation and hence provide stronger insight into how the rotation
changes with Galactocentric radius outside the Solar Circle. This particular
sightline is also one that presents relatively low total extinction (Sale et
al., 2014) and limited CO emission (Dame, Hartmann & Thaddeus, 2001), thereby
promising access to greater distances. It is consistent with these properties
that there is a raised stellar density visible at this location of up to twice
the average for the region (see Farnhill et al., 2016). This pencil beam most
likely misses the main Perseus spiral arm, since the main belt of extinction
and CO emission lies at higher latitudes than our chosen pointing by one to
two degrees. It is worth noting that the necessarily sparse H ii region data
used by Brand & Blitz (1993) to measure the outer-disc rotation curve,
revealed little change in apparent RV with distance in this location: all
values cluster around an LSR value of $-$50 km s-1. Russeil (2003) reported a
departure of $-21\pm 10.3$ km s-1 from mean circular speed associated with the
Perseus Arm.

  * •

Sightlines near the Galactic Anticentre intersect the direction of dominant
circular motion essentially at right angles, leading to the expectation of
measured RVs close to 0 (in the LSR frame) exhibiting negligible change with
increasing distance. We chose to observe an anticentre sightline so that it
serves as a control directly revealing e.g. RV measurement bias and the
magnitude of kinematic scatter. Again, the particular choice made is of a
pencil beam that is subject to relatively light extinction. At
$l=178^{\circ}$, higher extinction is found a degree away, on and below the
Galactic equator.

The targets we observed have magnitudes spanning the range, $14.2\leq i\leq
18.5$, which are expected to populate a heliocentric distance range of
approximately 2 to 10 kpc.

Figure 1: IPHAS colour-colour diagram with HectoSpec targets overlaid. The
grey points are IPHAS sources with $r<19$. The HectoSpec stars fall in two
distinct selection strips: the top strip selecting F-type stars (red points)
and the bottom strip selecting A-type stars (blue points). The black line is
an empirical unreddened main-sequence track, and the green dashed line is the
early-A reddening line.

The sample was selected using the IPHAS $r-i$, $r-H\alpha$ colour-colour
diagram, as shown in figure 1. Drew et al. (2008) described how this colour-
colour diagram can be used to effectively select samples of near main sequence
early-A stars. The candidate A stars were selected from a strip 0.04 mag wide,
just above the early A reddening line shown in figure 1, and the candidate F
stars were selected from a strip 0.08 - 0.09 mag above the same line. The use
of both F and A stars ensure the heliocentric distance distribution is well-
sampled, as the selected F stars will lie closer on average than the
intrinsically brighter A stars.

###  2.2 Observations and selection of data for analysis

The spectra were gathered using the MMT’s multi-object spectrograph, HectoSpec
(Fabricant et al., 2005), with the 600 lines mm-1 grating providing $\sim
2.4\,\mbox{\AA}$ resolution as measured from narrow sky lines. The
observations were performed over 6 nights in Sep-Nov 2011 and cover a
wavelength range of $6532-9045\,{\mbox{\AA}}$ with a sampling of
$0.56\,{\mbox{\AA}}$. Table 1 details the observations on each date, giving
the $i$ magnitude range targeted, exposure time, achieved count range, and
number of targets.

The HectoSpec data were reduced using the HSRED666Information on the latest
version of HSRED can be found at https://www.mmto.org/node/536 pipeline, and a
mean sky spectrum is determined using dedicated sky fibres. This is subtracted
off all target fibres in the same configuration. An example of a sky-
subtracted spectrum of an A star (blue line) and its corresponding sky
spectrum (orange line) can be seen in figure 2. The wavelength range shown
($8470-8940\,{\mbox{\AA}}$) is the region we use for measuring RVs and stellar
parameters. Whilst much smaller than the total range covered by the data, it
is chosen as it covers the CaT lines and some prominent Paschen lines. It is
also relatively unaffected by telluric absorption lines. The only other strong
photospheric feature potentially available to us is H$\alpha$, which is very
far away in wavelength, and close to the blue limit of the MMT spectra. The
region between the red dashed lines in figure 2 is excluded from further
analysis since it is particularly dominated by bands of telluric emission
which have proved difficult to subtract cleanly in many cases. The region also
excludes the diffuse interstellar band at $8620\,{\mbox{\AA}}$.

Figure 2: Close up of the CaT region in the spectrum of an A star from the HectoSpec data set (blue) and the corresponding sky spectrum (orange). The latter has been scaled down to 15$\%$ of the original counts. The region between the red dashed lines is excluded from the MCMC full-parameter fitting procedure since this region is affected by sky lines which frequently do not subtract well. The green line is the MCMC template fit. The narrow emission-like features in the star spectrum are incompletely-subtracted sky lines. The prominent $\sim$triangular absorption lines are Paschen lines, indicated by a circle. In later type stars, the CaT lines, indicated by a cross at 8498 Å, 8542 Å and 8662 Å become obvious, strengthening as the Paschen lines fade. Table 1: The details of the observations: date of observation, $i$ magnitude range targeted, exposure time, achieved count range, number of targets observed, Galactic longitude of sightline. Date of observation |  $i$ magnitude | Exposure (min) | Count range | Number of spectra |  $\ell$ (∘)
---|---|---|---|---|---
17-09-2011 | 16.5-17.5 | 140 | 390-5163 | 266 | 118∘
18-09-2011 | 16.5-17.5 | 135 | 1000-4012 | 259
19-09-2011 (a) |  $\leq$ 16.5 | 75 | 204-2983 | 258
19-09-2011 (b) | 17.5-18.5 | 265 | 143-2843 | 264
21-10-2011 |  $\leq$ 17.5 | 135 | 869-13003 | 256
18-11-2011 (a) |  $\leq$ 16.5 | 75 | 1452-12352 | 253
17-11-2011 |  $\leq$ 18 | 165 | 1192-4607 | 252 | 178∘
18-11-2011 (b) |  $\leq$ 17.5 | 120 | 662-41194 | 262

A quality cut was applied to the data, accepting spectra with an average count
level of more than 2000 (between $8475-8675\,{\mbox{\AA}}$), to ensure a large
enough signal-to-noise ratio (S/N) for reliable RV and stellar parameter
measurements. The wavelength-averaged minimum S/N corresponding to this is 23.
There are 887 spectra at $\ell=118^{\circ}$ and 434 spectra at
$\ell=178^{\circ}$ that survive this cut. Figure 3 shows the count levels vs
apparent $i$ magnitude, with the horizontal line representing the minimum
accepted count level. The distinct trends are due to weather, varying levels
of moonlight, and exposure time changes (see table 1). The configurations
exposed on 19-09-2011 and 18-11-2011, both for targets with $i\leq 16.5$,
present contrasting count levels due to a significant transparency change. The
effective magnitude faint limit is $i\sim 17.5-18$.

HectoSpec spectra that show signs of red-leak contamination are betrayed by an
upturn in the continuum at the red end reversing the decline seen shortward of
$\sim 8600\,\mbox{\AA}$. These spectra are removed from the sample, leaving
855 target spectra in the $\ell=118^{\circ}$ sightline and 409 in the
$\ell=178^{\circ}$ sightline.

Figure 3: Count level vs. apparent $i$ magnitude for $\ell=118^{\circ}$
(green) and $\ell=178^{\circ}$ (orange). The grey dashed line represents the
minimum accepted count level of 2000. There are 749 stars with counts $<2000$,
and 1321 with counts $\geq 2000$.

##  3 Analysis of spectra

###  3.1 Spectral fitting process

####  3.1.1 Overview

RVs and the stellar parameters (effective temperature, Teff, surface gravity,
$\log{g}$, and rotational velocity, v$\sin{i}$) were measured using Markov
Chain Monte Carlo (MCMC) full-parameter fits over the CaT range (8470-8940 Å).
The target spectra are compared with a template set, interpolated as needed,
by mapping the templates directly on to the observations and hence eliminating
the need for separate continuum fitting to both the template spectrum and
target spectrum.

The alternative method of RV measurement by cross-correlation was also
explored. Each target spectrum was cross-correlated with every template, and
the stellar parameters were adopted from the template which produced the
tallest cross-correlation function peak. However, a significant weakness of
this method is that it does not lend itself easily to error propagation.
Consequently, the MCMC method has been favoured. The RV measurements for both
methods are in agreement within the errors, with a slight bias to more
negative values in the case of cross-correlation. The median difference
between MCMC and cross-correlation RV measurements is 2.0 km s-1 – an amount
that falls inside the median RV error estimate of $\sim 4.4/6.8$ km s-1 for F
stars/A stars.

####  3.1.2 Template spectra

In order to determine the needed quantities from the target spectra obtained,
comparisons need to be made with a set of synthetic spectra. These synthetic
spectra were calculated using the approach of Gebran et al. (2016) and
Palacios et al. (2010). These authors have used SYNSPEC48 (Hubeny & Lanz,
1992) to calculate the spectra based on ATLAS9 model atmospheres (Kurucz,
1992), which assume local thermodynamic equilibrium, plane parallel geometry
and radiative and hydrostatic equilibrium. We collected 735 spectra that
sample the parameter domain as follows:

  * •

$[$Fe/H$]=0,-0.5$

  * •

$[\alpha$/Fe$]=0$

  * •

$5000\leq$ Teff (K) $\leq 15000$, in steps of $500$ K

  * •

$3.0\leq\log{g}\leq 5.0$, in steps of $0.5$

  * •

$0\leq$ v$\sin{i}$ (km s-1) $\leq 300$ km s-1, in steps of 50 km s-1

The metallicity is not treated as a free parameter: instead we use two
distinct template sets, one with $[$Fe/H$]=0$ and the other $[$Fe/H$]=-0.5$,
and compare the results. We do this because of the limited spectral coverage
and resolution of our data. Our numerical trials of metallicity as a free
parameter showed it to be underconstrained and prone to interfere with the
descent onto the values of other parameters. Nevertheless, there is an
expectation that with increasing heliocentric distance along both pencil
beams, there is likely to be a gradual decline in metallicity. Friel et al.
(2002); Chen, Hou & Wang (2003); Rolleston et al. (2000) found the metallicity
gradient in the Milky Way to be $\sim-0.06$ dex kpc-1. At the median distances
for the targets in each sightline, R${}_{G}\sim 11$ kpc for $\ell=118^{\circ}$
and R${}_{G}\sim 13$ kpc for $\ell=178^{\circ}$, this corresponds to
$[$Fe/H$]\sim-0.18$ and $[$Fe/H$]\sim-0.29$ respectively. In contrast, Twarog,
Ashman & Anthony-Twarog (1997); Yong, Carney & Teixera de Almeida (2005) found
$[$Fe/H$]=-0.5$ at R${}_{G}\geq$ 10 kpc. Hence it is appropriate to gauge the
effect of modest changes in adopted metallicity.

The model spectra have a resolution of R $=10000$, cover
$3600-9650\,{\mbox{\AA}}$ and have a constant wavelength sampling of
$0.05\,\mbox{\AA}$. However for this work, the templates have been broadened
with a gaussian filter to match the resolution of the target spectra and
rebinned to a sampling of $0.56\,{\mbox{\AA}}$ (also to match the data).

####  3.1.3 MCMC assisted full-parameter fitting

The parameters of astrophysical interest that we need to derive from each
spectrum are Teff, $\log{g}$, v$\sin{i}$ and RV. The first step in this
process has to be a mapping of template spectra on to each observed spectrum.
The mapping function we have adopted is linear which we find to be an adequate
approximation over the short wavelength range considered. This introduces two
extra free-parameters to be fit: the gradient and the intercept of the mapping
function. This approach is preferred over the alternative of separate
continuum fits to the template and observed spectra because it is then
possible to monitor how the mapping numerically influences the outcome.

The posterior probability distributions of the 6 free parameters (Teff,
$\log{g}$, v$\sin{i}$, RV, normalisation gradient and intercept) are obtained
assuming a likelihood function of the form:

$\mathcal{L}(\mu,\sigma^{2};x_{1},...,x_{n})\propto
e^{-\frac{1}{\alpha}\sum\limits_{i}{\frac{(x_{i}-\mu_{i})^{2}}{2\sigma_{i}^{2}}}}$
(1)

where $x$ refers to the target star spectrum sampled at pixels $i=1...n$ and
$\mu$ refers to the template spectrum. Since the pixels in a resolution
element are not independent, we compensate by dividing the summation by the
number of pixels per resolution element, $\alpha$. This is 4.3 in our case.
$\sigma_{i}$ is the noise level of the target star at pixel $i$, given by

$\sigma_{i}=\sqrt{\sigma_{sky+star,i}^{2}+\sigma_{sky,i}^{2}}$ (2)

where $\sigma_{sky+star,i}^{2}$ is the count level of the raw spectrum (the
sky and star together), and $\sigma_{sky,i}^{2}$ is the count level of the sky
spectrum. This likelihood function assumes the target spectrum has gaussian
errors.

By linearly interpolating the template grid, templates with intermediate
parameter values are produced and hence the model parameter space is
continuous. The RV range is sampled by shifting the wavelength axis of the
template according to the Doppler formula.

The ‘emcee’ python package (Foreman-Mackey et al., 2013) is used to execute
the MCMC parameter space exploration. Our procedure is set up for 200
‘walkers’. The priors used are flat for all parameters, with the range
available to each parameter matching the range in the template set. The range
of the RV prior covers all realistic values ($\pm 500$ km s-1). The slope and
intercept of the function mapping the template on to the target spectrum are
nuisance parameters, for which the range on the prior was determined by
experimentation with the dataset. After many steps the walkers converge (we
use 2000 steps with a 700 step burn-in) and the distribution of parameter
values returned by them define the posterior probability distribution of the
parameters. These are typically of the form of a 2D gaussian. The medians of
the marginalized distributions are adopted as the best estimates of parameter
values and the uncertainties are based on the 16th and 84th percentiles.
Figure 2 shows an example of a template (green line) with best-estimate
parameters for the observed spectrum (blue line).

###  3.2 Derived stellar parameters

####  3.2.1 Teff

The measured Teff distribution for both sightlines combined can be seen in
figure 4. The red bars represent the stars originally selected from the IPHAS
$r-i$, $r-H\alpha$ colour-colour diagram as F stars, and the blue bars
represent the A stars. 78$\%$ of the stars selected as candidate F stars have
measured Teff values of $6000-7500$ K (typical of F stars), and $81\%$ of the
candidate A stars have Teff values of $7500-10000$ K (typical of A stars). It
is necessary to point out that the template set spans temperatures inclusive
of G stars (T${}_{eff}\leq 6000$ K) and B stars (T${}_{eff}>10000$ K). Stars
measured as G stars make up only $2\%$ of the total sample and B stars $4\%$.
These are mainly stars that have contaminated the selected IPHAS $r-i$,
$r-H\alpha$ regions. Given the modest levels of contamination (e.g only 12% of
the inital A star selection turned out to be cooler than 7500 K), the method
of selection is shown to be very effective and practically viable. For
simplicity in the remainder of the paper, we label stars as either F or
A-type, with T${}_{eff}=7500$ K as the boundary dividing them.

The positive and negative errors in Teff as a function of Teff are shown in
the top panel of figure 5. The median of the positive and absolute values of
the negative errors combined is $\sim$ 150 K however the average error
increases with temperature and there are a number of targets that have large
asymmetric errors, betraying an unresolved fit dilemma. As temperature
increases, the Paschen line depths increase until T${}_{eff}\sim$ 9000 K,
thereafter they become more shallow again. This means there can be a
degeneracy where the Paschen line profiles of a lower temperature template are
similar to that of a higher temperature template, with the addition of line
broadening effects from the $\log{g}$ and v$\sin{i}$ parameter. This set of
stars, with large Teff errors ($>1000$ K), consist of only 85 of the 1261
targets, and are henceforth removed from the sample. These are represented by
empty circles in figure 5.

Finally, this leaves a total sample made up of 705 A stars (T${}_{eff}>$ 7500
K) and 471 F stars (T${}_{eff}\leq 7500$ K): broken down into the sightlines,
there are 473 A stars and 310 F stars in the $\ell=118^{\circ}$ sample, and
232 A stars and 161 F stars in the $\ell=178^{\circ}$ sample.

Figure 4: Superposed distributions of measured Teff values (for both sight
lines combined) of stars selected as probably F-type (red histogram) and
A-type (blue histogram). Of the stars initially selected as A-type, only 12%
are cooler F/G stars.  Figure 5: Positive (red) and negative (blue) errors in
Teff (top panel), $\log{g}$ (middle panel) and RV (bottom panel) as a function
of Teff. The grey dashed lines in the top panel represent the Teff error cut:
points with $|error|>$1000 K that are removed from the sample are shown in
this figure, represented by empty circles. The green bars connect the positive
and negative errors for each individual target.

####  3.2.2 $\log{g}$

The middle panel of figure 5 shows the positive and negative errors in
$\log{g}$ from the posterior distributions as a function of Teff, for the A
and F stars at solar metallicity. It is clear from the figure that the formal
$\log{g}$ error rises steadily with decreasing effective temperature. The
median error at T${}_{eff}>7500$ K is 0.09, while below this it increases to
0.14. This trend most likely tracks the growing importance of H- continuum
opacity with decreasing effective temperature, causing the wings of the CaT
lines to become less sensitive to surface gravity. Gray & Corbally (2009) have
noted a ‘dead zone’ among mid-late F stars in which dwarf and giant spectra
are nearly indistinguishable.

In F stars with appreciably reduced Paschen line profiles, in particular, this
underlying astrophysical trend is compounded to an extent by the moderate
spectral resolution of the data. We find for these cooler stars that the fits
begin to exhibit a 3-way degeneracy for combinations of temperature, gravity
and metallicity. An example of this degeneracy at work in the CaT in cooler F
stars was presented by Smith & Drake (1987). However, with metallicity fixed,
it is possible to identify temperature and gravity albeit with greater error
on the latter. The returned F star $\log{g}$ distribution is skewed strongly
in favour of near-MS objects, with a median value of 4.5 and interquartile
range 4.1-4.8, tapering off into a tail reaching down to one object with $\log
g\simeq 3.0$ (the lower bound on the template set).

The distribution of best-fit $\log{g}$ values for the A stars can be seen in
figure 6. The peak of the distribution lies close to where we would expect it
to be. Also shown is the distribution of $\log{g}$ for A stars from a Besançon
model (Robin et al., 2003) for $\ell=118^{\circ}$, after convolving with a
gaussian of $\sigma=0.09$ to emulate the HectoSpec measurement error, for a
more useful comparison. The measured HectoSpec distribution has a tail at
large values of $\log{g}$ that is not seen in the Besançon model, and its peak
is not perfectly matched. A shift of the Besançon distribution of $+0.15$
brings the two distributions into rough alignment. This could be evidence of a
bias in our measurements. We appraise the impact of this in section 5.2. The
stars occupying the high-end tail, which appears more extended than in the
Besançon model distribution, are mainly cooler objects carrying larger-than-
median errors (see figure 5).

Figure 6: Distribution of measured $\log{g}$ values of A stars for both sight
lines combined (blue histogram). The distribution of $\log{g}$ values of A
stars from a Besançon model for $\ell=118^{\circ}$ is also shown (green
histogram). This has been convolved with a gaussian of $\sigma=0.09$ to
emulate the HectoSpec measurement error.

####  3.2.3 v$\sin{i}$

Figure 7 shows the measured v$\sin{i}$ distribution for both sightlines
combined, separated into F stars (red bars) and A stars (blue bars). The
distribution is as expected, with generally low values for F stars and a
spread from low to high values for A stars (Royer, 2014). The median error on
v$\sin{i}$ is $\sim 20$ km s-1, increasing to $\sim$40 km s-1 for
T${}_{eff}>10000$ K - as expected for stars that are more commonly fast
rotators. Unsurprisingly, there is evidence of a slight negative correlation
between individual $\log{g}$ and v$\sin{i}$ parameter fits.

Figure 7: The distributions of measured v $\sin{i}$ values, separated into F
stars (red) and A stars (blue), and shown overplotted. The two lines of sight
are merged within each distribution.

####  3.2.4 RV

Figure 5 shows the errors on RV as a function of Teff. The median error is
$\sim 4.4\,$km s-1 for F stars and $\sim 6.8$ km s-1 for A stars. This
difference is attributable to the growing contrast of the CaT lines with
decreasing effective temperature. The Solar motion needed to convert from the
heliocentric to the LSR frame is taken as $(U,V,W)=(-11,+12,+7)$ km s-1
(Schönrich, Binney & Dehnen, 2010).

###  3.3 A comparison with higher-resolution long-slit spectra

As a check on the reliability of the derived parameters and their errors, we
obtained additional long-slit red and blue spectra for 7 HectoSpec targets –
the positions and apparent $i$ magnitudes of which are shown in table 2. These
were accompanied by 5 RV standard stars (with 3 observed more than once). Of
particular interest in this comparison are the measured surface gravities and
RVs. The 7 objects reobserved are spread in $\log{g}$ from 3.5 to 4.5, as
determined from the HectoSpec data, lying across the peak of the distribution
in figure 6. The Teff range, again determined from the HectoSpec data, spans
$\sim 7900-9500$ K, with one object outside this range with $\sim
11900\pm_{2600}^{150}$ K – the large error indicating the unresolved fit
dilemma described in section 3.2.1.

Table 2: Positions and apparent $i$ magnitudes of the objects used for the HectoSpec-ISIS comparison. Target | RA | DEC |  $i$ mag | ISIS red Teff (K)
---|---|---|---|---
1 | 00:03:41 | 64:29:43 | 14.94 | 7665$\pm_{101}^{115}$
2 | 00:06:36 | 64:22:38 | 14.89 | 7703$\pm_{73}^{76}$
3 | 00:03:47 | 64:17:12 | 14.77 | 7769$\pm_{95}^{89}$
4 | 00:07:37 | 64:07:51 | 14.78 | 8331$\pm_{219}^{2957}$
5 | 00:05:36 | 63:57:55 | 14.84 | 11043$\pm_{1332}^{344}$
6 | 00:04:02 | 64:29:49 | 14.91 | 11299$\pm_{1710}^{158}$
7 | 00:07:13 | 64:46:18 | 14.82 | 11370$\pm_{378}^{202}$

These spectra were gathered as service observations using the Intermediate-
dispersion Spectrograph and Imaging System (ISIS) of the 4.2 m William
Herschel Telescope during 3 nights in Oct-Dec 2016. ISIS is a high-efficiency,
double-armed, medium-resolution (8-121 Å/mm) spectrograph. The R1200B grating
was used on the blue arm, and the R1200R grating on the red arm, providing
resolution elements for a 1” slit of 0.85 Å and 0.75 Å respectively. The
wavelength coverage of the blue spectra is 3800-4740 Å and of the red spectra
is 8110-9120 Å. Both the blue and red spectra have a constant wavelength
sampling of 0.22 Å and 0.24 Å respectively.

The raw images were processed and sky-subtracted and the resultant spectra was
wavelength calibrated, all with use of the Image Reduction and Analysis
Facility (iraf). The spectra were then passed through the MCMC full-parameter
fitting method. For the red spectra the same wavelength limits were adopted as
for HectoSpec, and for the blue the wavelength range used was
$4000-4600\mbox{\AA}$, covering some of the Balmer series whilst excluding the
Ca ii K and H lines since they suffer from interstellar absorption. The
measured parameters and RV were then compared to those measured from the
HectoSpec spectra, or to values from the literature in the case of the RV
standards, to reveal any systematic differences. A method check was performed
by comparing the measured parameters from the red and blue ISIS spectra,
eliminating the potential for differences due to a change of instrument. The
weighted mean difference between measured RV for the ISIS RV standard stars
and RV from the literature is $-0.7$ km s-1 for the blue spectra and $+0.1$ km
s-1 for the red spectra. Hence we are confident the ISIS wavelength scale is
reliable.

Figure 8 shows the difference between the ISIS measurements (blue points for
blue spectra and red points for red spectra) and the HectoSpec measurements.
The targets are in ascending order of Teff, as determined from the ISIS red
spectra. In general the differences in outcome are reassuringly modest and
show the method is working satisfactorily. The dashed lines represent the
weighted mean difference between the ISIS red and HectoSpec measurements (red
line) and ISIS blue and HectoSpec measurements (blue line). The weighted mean
difference between the red ISIS and HectoSpec measurements, and the standard
deviation of the spread, are: $\Delta$T${}_{eff}=-215\pm 1335$ K,
$\Delta\log{g}=-0.36\pm 0.18$, $\Delta$v$\sin{i}=-24\pm 53$ km s-1, $\Delta$RV
$=-1.9\pm 3.5$ km s-1. The small mean difference in measured RV indicates the
HectoSpec wavelength calibration is systematically offset by an amount well
below measured random errors. The larger difference in $\log{g}$ suggests
there may be a bias towards larger values in the HectoSpec data. This
potential bias is not the first evidence: the comparison between our A star
$\log{g}$ distribution and that from a Besançon model is shown in section
3.2.2 and we see the HectoSpec distribution peaks at a slightly larger value.
We return to the impact of this possible bias in section 5.2.

The blue data points in figure 8 provide the comparison between the ISIS blue
and HectoSpec data. The weighted mean offsets and standard deviations obtained
are: $\Delta$T${}_{eff}=-289\pm 1017$ K, $\Delta\log{g}=-0.43\pm 0.32$,
$\Delta$v$\sin{i}=-18\pm 63$ km s-1, $\Delta$RV $=-7.7\pm 8.8$ km s-1. As with
the red, there is a sizable offset in $\log{g}$, suggesting the HectoSpec data
may carry a positive bias. The RV difference is larger than that measured for
the red spectra, potentially as a result of the fewer absorption lines fitted
in the blue region and hence lower accuracy.

We have also reviewed the differences between parameters derived from the ISIS
blue measurements and the ISIS red and HectoSpec data. Hales et al. (2009)
performed a similar comparison between the red and blue ranges as observed
with ISIS, and found a tendency for the red determined spectral type to be
earlier by 0.9 subtypes. They attributed this to strengthening of the Ca ii H
and K absorption lines by an interstellar component. Although we exclude these
lines in the fitting procedure, we find a modest offset in effective
temperature, with the red suggesting a slightly hotter star of 112 K with very
little change in surface gravity (0.02 dex).

The comparison with higher-resolution spectra has provided useful insights –
the offsets between the ISIS and Hectospec measurements are reassuringly
modest and are similar to those expected due to random errors. Our employed
method is working well. The larger offset in the derived surface gravities
suggest a bias could be present to which we will continue to pay attention.

Figure 8: Differences between ISIS and HectoSpec measured stellar parameters.
The blue points represent the differences between measurements from the blue
ISIS and HectoSpec spectra, and the red points specify differences between the
red ISIS and HectoSpec spectra. The targets are in ascending order of Teff, as
determined from the ISIS red spectra. The error shown on each datapoint is the
quadratic sum of the HectoSpec error and ISIS error. The dashed lines
represent the weighted mean difference between the ISIS red and HectoSpec
measurements (red line) and ISIS blue and HectoSpec measurements (blue line).

###  3.4 Extinction, absolute magnitudes and distance modulii

The distance modulus, $\mu$, of each star is calculated via the following
equation

$\mu=m_{i}-A_{i}-M_{i}$ (3)

where mi is the apparent magnitude in the $i$ band, Ai is the extinction and
$M_{i}$ is the absolute magnitude. The absolute magnitudes used are from
Padova isochrones, interpolated with Teff and $\log{g}$ scales (Bressan et
al., 2012; Chen et al., 2015). We choose a value for $M_{i}$ on the basis of
the median $\log{g}$ and Teff returned by the fits. Where $\log{g}$ exceeds
the maximum present in the Padova isochrones for the specified Teff, we reset
$\log{g}$ to this value. The median error on the absolute magnitudes (due to
stellar parameter uncertainties) is $\sim$0.3. This is the dominant error
source in our final results.

The extinction of each target was calculated using

$A_{i}=2.5[(r-i)_{obs}-(r-i)_{int}]$ (4)

where $(r-i)_{obs}$ is the observed colour of the star, taken from IPHAS
photometry, and $(r-i)_{int}$ is the intrinsic colour of the star. The
intrinsic colours are calculated from the template grid via synthetic
photometry and the value for each target star is interpolated on this grid
based on the measured Teff and $\log{g}$. The coefficient of 2.5 is the ratio
of Ai to A${}_{r}-$Ai for main-sequence A/F stars with reddening levels
similar to the HectoSpec data. We use the Fitzpatrick law with R${}_{V}=3.1$.
The median error on intrinsic colour (due to stellar parameter uncertainties)
is $\sim$0.01, and on Ai (due to stellar parameter and photometric
uncertainties) is $\sim$0.05. After dereddening the observed magnitudes, $\mu$
is obtained: across the entire sample the interquartile range for the error in
$\mu$ is 0.2 to 0.4. The smallest errors are associated with early A stars
(median error $\sim 0.2$).

Figure 9 shows the Ai extinctions as a function of estimated distance modulus
for both sightlines. Also shown are the mean extinction trends due to Sale et
al. (2014) across the two pencil beams, including the expected dispersion in
extinction (grey-shaded region). This comparison provides some insight into
the plausibility of the distance modulus distribution of our sample. An
important difference to be aware of is that the Sale et al. (2014) trends were
computed from IPHAS photometry of all probable A-K stars, down to apparent
magnitudes that are appreciably fainter (to $i\sim 20$) than those typical of
our analysed spectra ($i<18$). In both sight lines, there is evidence that the
spectroscopic samples favour lower extinctions than the fainter-weighted
photometrically-based trends. This is most likely a straight-forward selection
effect.

To check this, we have examined the impact of the counts cut we placed on the
spectra included in the analysis: specifically, we have compared median
estimates of extinction, derived from their $r-i$ colour, for the stars
analysed (counts $>2000$), with those for lower-count stars not analysed. We
find that for both directions the cut biases the typical extinction of the A
stars (inferred from the available $r-i$ colours) to lower values: the
strength of effect is that the unanalysed objects have a median Ai that is
greater by $0.4$ magnitudes. This helps explain the tendency for the blue
A-star data points, particularly, to sit lower in figure 9. Among the cooler,
on-the-whole closer, F stars there is little difference. The alternative
explanation for this – overlarge distance moduli – runs into difficulty when
it is recalled that there may be systematic overestimation of surface
gravities (see sections 3.2.2 and 3.3). We view this comparison as tensioning
against accepting and correcting for such a bias in $\log{g}$, as this would
drive up the distance moduli, creating a yet bigger offset.

Figure 9: The extinction, Ai, of F stars (red points) and A stars (blue
points) as a function of distance modulus. Top: $\ell=118^{\circ}$, bottom:
$\ell=178^{\circ}$. Also shown in each panel are the photometrically-predicted
mean extinction trends (grey lines) across the pencil beam (Sale et al.,
2014). The grey shaded region defines the expected dispersion in extinction at
every distance within the beam. Stars that lie far from the trend in the upper
panel are highlighted by a change of colour – orange for the F stars and cyan
for one A star. This colouring is preserved in figure 11.

###  3.5 The effect of metallicity

So far all the parameters obtained and described have been computed for solar
metallicity.

Stellar parameters and RVs have been measured again, as described in section
3.1.3, but with $[$Fe/H$]$ set to -0.5 for the A stars. A new Padova absolute
magnitude scale, suitable for the changed metallicity, was used for
determining distance moduli. The reduced-metallicity fits typically return
cooler temperatures ($\Delta$T${}_{eff}\sim-400$ K). The $\log{g}$ values are
also lower ($\Delta\log{g}\sim-0.17$), partly compensating for the cooler
temperatures in the estimation of distance. The net effect on the distance
modulus scale is a small increase compared with the solar metallicity scale
($\Delta\mu\sim+0.12$). The RVs measured adopting $[$Fe/H$]=-0.5$ are slightly
more negative: the median difference is $\sim-3.0$ km s-1.

In the case of the F stars, there is a growing degeneracy between Teff,
$\log{g}$ and metallicity (see section 3.2.2). In a trial of $[$Fe/H$]=-0.5$
in fitting the F stars, we found the returned gravities to be unrealistically
low for a population of objects that is more localised than the A stars whilst
sharing the same faint magnitude limit. Moreover, it is highly improbable that
many among the F star sample would present $[$Fe/H$]$ significantly less than
0, since the great majority of these fainter objects should lie within a
distance of 5 kpc. We estimate this limiting radius for the case of a warmer
main sequence F star with T${}_{eff}=7000$ K, $i=17.5$ (see figure 3),
$A_{i}\sim 1.3$ (typical HectoSpec value – see figure 9) and $M_{i}=2.6$. At
$\ell=118^{\circ}$, a distance of 5 kpc corresponds to a Galactocentric radius
of $\sim$10.6 kpc, or a metallicity change of $\sim 0.2$. Consequently
throughout this paper we will fix the metallicity of the HectoSpec F stars at
$[$Fe/H$]=0$.

##  4 The sightline radial velocity trends

The trend of RV with distance modulus can be compared with what we would
expect to see as an averaged trend if consensus views of the Galactic rotation
law apply. If all the stars move on circular orbits about the Galactic centre,
we would expect to observe a trend of RVs that are a spread (due to velocity
dispersion) around a curve whose shape depends on the sightline observed, the
assumed rotation curve and LSR parameters (circular velocity and
Galactocentric distance of the LSR, V0 and R0). The comparisons made in this
section are with a flat rotation curve with LSR parameters R${}_{0}=8.3$ kpc,
V${}_{0}=240$ km s-1, and with the slowly rising rotation curve derived in
Brand & Blitz (1993) with their adopted parameters R${}_{0}=8.5$ kpc,
V${}_{0}=220$ km s-1. We begin with the results for $\ell=178^{\circ}$, the
control direction, where the rotational component of motion of the stars is
effectively nullified.

###  4.1 $\ell=178^{\circ}$

The trend of RV with $\mu$ for this pencil beam is shown in figure 10. The red
dashed line shows the expected average trend based on a flat rotation curve
(although at this sightline the shape of the rotation curve makes little
difference). The green line is the weighted mean of the RVs for the solar
metallicity stars, plotted as individual data points, at each step in $\mu$.
The yellow line is the weighted mean for A stars with $[$Fe/H$]=-0.5$ and F
stars with solar metallicity. The weighted RV is calculated via the following
process:

  * •

Every ($\mu$, RV) data point is assigned a normalised gaussian error
distribution with $\sigma=\sigma_{\mu}$ where $\sigma_{\mu}$ is the error on
$\mu$.

  * •

At the ith step ($\mu=\mu_{i}$), the data points ($\mu_{n}$, RVn) with error
distributions in $\mu$ overlapping $\mu_{i}$ are included in forming the
running average, provided $|\mu_{n}-\mu_{i}|<\sigma_{\mu_{n}}$. The weighting
per data point is in proportion to the value of its gaussian error function at
$\mu_{i}$ – thereby taking into account the error in $\mu$ in all data points.

  * •

In obtaining the mean RV at $\mu=\mu_{i}$, the contributing RV values are also
weighted in proportion to their errors.

  * •

A final weighting is applied in computing the mean in order that the
contribution from data points at $\mu<\mu_{i}$ balances the contribution from
data points at $\mu>\mu_{i}$. This limits the influence of data from the most
densely populated part of the $\mu$ distribution on the mean trend.

  * •

The minimum number of points required for calculation of the mean to either
side of $\mu=\mu_{i}$ is 50.

The shaded region around the mean trend line represents its error – the
standard error of mean RV at each $\mu_{i}$.

Figure 10: The trend of RVs with distance modulus for $\ell=178^{\circ}$. The
green line is the weighted mean of the RVs of the solar metallicity stars, and
the yellow line is obtained on combining $[$Fe/H$]=-0.5$ A stars with solar
metallicity F stars. The shaded regions represents the standard error of the
mean in RV. The blue points are A stars and the red points F stars, both with
metallicity assumed as solar. The red dashed line shows the expected trend for
a flat rotation law, which at this sightline will fall close to $\sim$zero
regardless of the V0, R0 or shape of the rotation law adopted.

In figure 10, the F stars dominate the lower end of the range in $\mu$, while
the A stars mostly occupy a spread from $\mu\sim 12$ to $\mu\sim 15$ (or a
heliocentric distance range from 2 to 10 kpc). Of the F stars, 90$\%$ (145 of
165) lie within $\mu=11.4-14.5$ (d$=1.9-7.8$ kpc). 90$\%$ of the A stars with
$[$Fe/H$]=0$ (209 of 232) lie within $\mu=11.8-14.8$ (d$=2.3-9.0$ kpc), and
90$\%$ of the A stars with $[$Fe/H$]=-0.5$ (130 of 144) lie within
$\mu=11.6-15.1$ (d$=2.1-10.6$ kpc). Ruphy et al. (1996) found a radial cutoff
of the stellar disc at R${}_{G}=15\pm 2$ kpc, and Sale et al. (2010) found the
stellar density of young stars declines exponentially out to a truncation
radius of R${}_{G}=13\pm 0.5_{statistical}\pm 0.6_{systematic}$ kpc, after
which the stellar density declines more sharply. The density of our sample in
this sightline also drops off at these distances, but at this stage we cannot
be sure whether this decline originates in the Galactic disc or is a selection
effect.

We find the overall trend in RV follows the expected flat behaviour, although
it is offset in velocity by a mean amount of $3.5$ km s-1. The results
adopting $[$Fe/H$]=-0.5$ for the A stars are very similar as expected, but
with a slightly smaller mean offset of $3.0$ km s-1. There was evidence of a
possible small wavelength calibration offset arising from the comparison of
the HectoSpec observations with independent higher-resolution spectra,
effecting the RV scale by only a couple of km s-1, as described in section
3.3. Taking these findings together we infer that the RV scale is reliable to
within $\sim 5$ km s-1 but cannot rule out the presence of a small positive
bias in the measurements.

We have computed the RV dispersion of the sample stars, in broad spectral type
groups, around the measured mean trend, in order to compare them with the
expected dispersions from Dehnen & Binney (1998) (see table 3). Clearly the
measured dispersions are larger than the Dehnen & Binney (1998) results. Part
of this discrepancy is attributable to our measurement errors, but the larger
share of it may be a consequence of stellar multiplicity. At least a half of
the sample objects are likely to be members of multiple systems, and of those
around 15% may be in nearly equal mass ratio binaries (Duchêne & Kraus, 2013).
Whilst the spectroscopic resolution of our data is insufficient to pick out
spectroscopic binaries, there can nevertheless be binary orbital motions
present up to a level of $\sim 40$ km s-1 (depending on phase) in typical
cases (for total system masses of $3M_{\odot}$, with period P $\sim 40$ days).
As the data in table 3 indicate, the excess dispersion on top of measurement
error is in the region of $10$ km s-1 for all 3 spectral type groups. We look
at this in more detail in section 5.

Table 3: Expected RV dispersion from Dehnen & Binney (1998) ($\sigma_{DB}$) compared to the measured RV dispersion ($\sigma_{178^{\circ}}$) for early A - early F stars (solar metallicity set). Also given are the typical HectoSpec measurement errors ($\epsilon_{RV}$) for the different subtype ranges, and the quadrature sum in combination with $\sigma_{DB}$. Finally, the excess $\sigma$ required to reconcile $\sigma_{DB}$ and $\sigma_{178^{\circ}}$ are provided. These excesses are compatible with the extra dispersion likely to be introduced by the presence of spectroscopic binaries in the sample (on the order of $10$ km s-1). Stellar Type | Teff range | No. stars | $\sigma_{DB}$ | $\epsilon_{RV}$ | $\sqrt{\sigma^{2}_{DB}+\epsilon^{2}_{RV}}$ | $\sigma_{178^{\circ}}$ | excess $\sigma$
---|---|---|---|---|---|---|---
Early A | ]8500-10000] | 67 | 14 | 7.2 | 15.7 | 20.7 | 13.5
Late A | ]7500-8500] | 151 | 19 | 5.5 | 19.8 | 21.7 | 8.9
Early F | ]6500-7500] | 139 | 23 | 4.2 | 23.4 | 27.8 | 15.0

###  4.2 $\ell=118^{\circ}$

The results for this sightline are shown in figure 11. Again, the green line
is the mean trend for solar metallicity stars and the yellow for
$[$Fe/H$]=-0.5$ A stars and solar metallicity F stars. The mean trends,
calculated in the same way as for $\ell=178^{\circ}$ described in section 4.1,
deviate more strongly from expectations based on a flat or slowly rising
rotation curve (shown in the figures by the red dashed/dotted lines). The
measured trends are appreciably flatter. The $\ell=118^{\circ}$ results are
similar for both A star metallicities, indicating that metallicity is not
having a strong effect on the interpretation of the results. The mean RV trend
(for either A-star metallicity) spans the heliocentric distance range from
$2.5$ to $9$ kpc (R${}_{G}\simeq 9.7$ to $14.8$ kpc), with the RV decreasing
slowly from $\sim-40$ to $-70$ km s-1.

For context, we note that the Perseus Arm is likely to pass closest to this
line of sight at a distance of 2.5 to 3 kpc (Reid et al., 2014). This overlaps
the bottom end of the distance range that we sample. In figure 11 there is no
clear sign of a distinct localised RV perturbation that might be attributed to
the arm. Spiral arm perturbations are discussed further in section 5.3.

The velocity dispersions around the RV mean trend for the three spectral type
groups specified in table 3 at the $\ell=118^{\circ}$ sightline are: 19.8 km
s-1 for the early A stars, 21.1 km s-1 for the late A stars, and 24.3 km s-1
for the early F stars. These values are similar to those measured for the
anticentre sightline (shown in table 3), which permits the same interpretation
that binary orbital motions are a third factor contributing to the overall
observed dispersion.

In the next section, we consider whether the photometric consequences of
binarity could distort the results, discuss the observed departure from a flat
or slowly rising rotation law, and consider the possibilities for detecting
spiral arm perturbations.

Figure 11: The trend in RV with distance modulus at $\ell=118^{\circ}$. The
red dashed/dotted lines show the expected trends for a flat rotation law and
the Brand & Blitz (1993) slowly rising rotation law (see legend). The green
line is the weighted mean of the RVs obtained using stellar parameters
returned for solar metallicities, and the yellow line is the alternative
result obtained on swapping in A star data computed for $[$Fe/H$]=-0.5$. The
shaded regions represent the standard error of the mean in RV. The individual
data points in blue are obtained from A stars, while red is used for the F
stars (computed for solar metallicity). The cyan point is the A star
identified as lying far from the IPHAS mean extinction trend, and the orange
points the F stars (see figure 9).

##  5 Discussion

###  5.1 Potential bias from binaries and distance error

The relatively flat distribution of RVs seen at $\ell=118^{\circ}$ in place of
the expected falling trend may possibly be a consequence of unrecognised
binaries placed at inappropriately short distances, or of significant smearing
due to distance error.

If a target star is actually an unresolved binary system, the measured
apparent magnitude will be brighter than if it were a single star. In effect,
the absolute magnitude adopted in our analysis is then too low, and hence the
distance will be underestimated. Similarly, if the component masses are
unequal, the colour will be redder than if the system were a single star,
resulting in overestimation of the extinction and further underestimation of
distance. The additional velocity component from the motion around another
star in a binary system does not produce a bias on the RV results since the
space orientation of binary orbital axes across a large sample is random and
must cancel out. The photometric effect will be most pronounced in nearly
equal mass binary systems, and has been quantified by Hurley & Tout (1998).
Since the binary mass ratio distribution in A/F stars is not far from flat, we
might expect of order $10-20\%$ of our sample to be misconstrued as closer to
the Sun by up to $0.75$ in $\mu$. In principle this might erroneously flatten
the overall RV distribution by bringing in a group of objects at more negative
RV to mix with less negative values at smaller $\mu$.

In order to test the effect of binarity on the calculated weighted average RV
trend, we have performed an outline simulation that focuses on this factor.
Our method is as follows. We select three sets of stars of different spectral
type groups: early A, late A and early F. The size of the sets are the same as
the HectoSpec $\ell=118^{\circ}$ groups (early A: 168 stars, late A: 281
stars, early F: 259 stars). 70% of stars in each group are randomly selected
as binaries – these are assigned a primary mass, $m_{1}$, and the secondary
mass, $m_{2}$, is assigned at random according to a mass ratio fraction, q,
obeying the distribution $f(q)=q^{-0.5}$ (Duchêne & Kraus, 2013). The distance
modulus assigned to each star is sampled from the HectoSpec $\mu$ distribution
for its spectral type group. In order to compensate for the expected net
decrease in $\mu$ associated with treating output binary stars as single,
these reference distributions are first modified by a uniform retrospectively
determined shift of +0.16. The assigned RVs follow a flat rotation curve with
V${}_{0}=240$ km s-1 at $\ell=118^{\circ}$, broadened by an amount consistent
with the scatter defined by Dehnen & Binney (1998) and measurement error. The
final adjustment is to include a component of binary orbital motion. To
achieve this, a period, P, inclination, i, and phase, $\phi$, must be randomly
assigned for each binary star. The period is selected from the distribution of
A star periods shown in figure 2 in Duchêne & Kraus (2013). The inclination is
chosen from a uniform distribution in $\cos{i}$, and the phase from a uniform
distribution between $0$ and $2\pi$. In the final reconstruction of the
observed RV distribution, each simulated binary star has its distance modulus
reduced to the equivalent single value according to the computed difference in
intrinsic colour and absolute magnitude.

The simulation of 708 stars was performed 10000 times, and the mean RV trend
was calculated each time. The mean of these 10000 trends is shown as the
purple line in figure 12. The binary stars are pulled by varying amounts to
shorter distances and bring with them their on-average more negative RV. A
flattening of trend is seen, but our numerical experiment indicates it is
modest. The deviation away from a flat rotation law we find at
$\ell=118^{\circ}$ (figure 11) is much more pronounced, leading us to conclude
that an appeal to stellar multiplicity to explain it falls well short,
quantitatively.

Another factor that will cause some flattening of the $\ell=118^{\circ}$ trend
is distance error. The derived distance distibution is essentially the true
distance distribution broadened by the uncertainty. Consequently the mean RV
trend spans this slightly broader distribution, causing a flattening. To test
the extent of this we performed a simulation similar to the one described
above used to test the effect of unresolved binaries. However this time we
disregard binary stars, and after assigning a suitably scattered RV with
measurement error to each notional star, we shifted the corresponding distance
modulus by an amount within the typical level of error from the HectoSpec
data. The result is shown as the orange line in figure 12. The resultant mean
trend is somewhat flattened, but not by an amount that makes it parallel to
the result from observation. To complete the picture, we performed this
simulation again but now including unresolved binaries as described above. The
blue line in figure 12 shows the result. The amount of flattening is similar
to that due to the distance errors alone. At shorter distances it is slightly
more pronounced, and at further distances it is less. This is expected since
the binaries are pulled to shorter distances, bringing with them their on-
average more negative RV.

In order to achieve a flattening of the simulated curve by an amount that
begins to mimic the trend deduced from the A/F star data, we find we need to
double the distance errors relative to those propagated from the data as
described in section 3.4. The second panel of figure 12 illustrates this.
Whilst it is certainly a possibility the estimation of distance errors in our
sample is optimistic at $\sim$ 15% ($\Delta\mu=0.3$), growing them all by as
much as a factor of two is rather less credible.

Figure 12: The results of simulations to test the effects on the mean RV trend
of undetected binaries and distance modulus errors. In the top panel, the
purple line is the mean RV trend for the simulation incorporating unresolved
binaries. The orange line is the result from the simulation testing the effect
of distance modulus error only, while the blue line shows the simulated effect
of both unresolved binaries and distance modulus errors. The observed RV trend
is overlaid in green. In the bottom panel, we test the effect of doubling our
distance modulus errors (orange line). The greater error induces more
flattening. The light green line in this case is the mean RV trend of the
HectoSpec data obtained on adopting doubled distance modulus errors.

###  5.2 Comparisons of results with other tracers and earlier work

Using H ii region data, Brand & Blitz (1993) presented sparse-sampled
measurements that mainly captured heliocentric distances out to $\sim$4 kpc at
the Galactic longitudes of interest here (see their figure 1). The overlap
with our results thus runs roughly from 2 to 4 kpc. Near the anticentre Brand
& Blitz (1993) generally favoured slightly negative radial velocities, ranging
from $-18$ to $+8$ km s-1 (9 datapoints, from their table 1), to be compared
with a small positive bias here (figure 10). In the longitude range
$110^{\circ}<\ell<130^{\circ}$, the relevant measurements are spread between
$-30$ and $-56$ km s-1 (13 datapoints). This is entirely compatible with our
results.

A denser comparison between our results and other studies can be made using RV
data of H i and CO clouds. Since both sightlines miss the latitude of peak
gaseous emission for their longitudes, the total amount of both H i and CO is
not particularly large. Nevertheless, the measured gaseous RV distributions
are broadly consistent with our findings from HectoSpec, and the detail is
informative. H i 21 cm data from EBHIS (The Effelsberg-Bonn H i Survey, Winkel
et al., 2016) in the $\ell=178^{\circ}$ sightline scatter around RV $\simeq$ 0
at much reduced dispersion compared to the A/F stars – as expected. We find
that the mean RV measures from our optical spectroscopy are shifted relative
to the H i data by $\sim+8$ km s-1, and regard this mainly as evidence that
the H i column samples a greater column through the outer disc than our
stellar sample.

The comparison with EBHIS data at $\ell=118^{\circ}$ shown in figure 13 shows
the A/F star data line up with the main $\sim-60$ km s-1 H i emission peak,
while the peak at $\sim 0$ km s-1 (from the Local Arm, seen in H i) is clearly
absent. This is to be expected given that none of the A/F stars selected and
measured will be in or near the Local Arm. The H i data also present a peak at
$\sim-90$ km s-1 that is largely absent from the stellar data. This is
unsurprising since our central result from the $\ell=118^{\circ}$ sightline is
the RV flattening that implies a relative absence of stars in this more
negative velocity range (at distances where a flat or gently rising rotation
law would predict they exist). A reasonable inference from this is that the H
i gas at the most negative radial velocities lies mainly outside the range
sampled by the A/F stars. We note that the CO data from the COMPLETE
(Coordinated Molecular Probe Line Extinction and Thermal Emission Ridge et
al., 2006) survey exhibit the same RV peaks as the H i data. This difference
either indicates that the H i and CO gas lies beyond $d\sim 9$ kpc, or that
the distance range occupied by our sample of stars does not extend to the
distances where existing rotation laws would predict RV$\sim-90$ km s-1. From
figure 11, it can be seen that these laws associate a distance of $\sim 8$ kpc
with this RV. We argue below the stars in our sample placed at $\sim 8$ kpc
are not there simple because of distance error.

Figure 13: The H i profile at $\ell=118^{\circ}$ (orange line), overlayed on
the HectoSpec RV data (green histogram).

Huang et al. (2016) have used red clump giants drawn from a wide range of
Galactic longitudes sampling the outer disc to find a broadly flat longitude-
averaged rotation law within R${}_{G}<25$ kpc, with typical circular speed
V${}_{0}=240\pm 6$ km s-1. But over our sampled region, between R${}_{G}\simeq
10-15$ kpc, their inferred rotation law is quite sharply rising out of a dip
at R${}_{G}\sim 11$ kpc. Figure 14 compares the Huang et al. (2016) results
(red line) with the rotation curve derived from the mean RV trend we obtain at
$\ell=118^{\circ}$ adopting solar metallicity (green line). In constructing
this figure, we choose the same LSR parameters as favoured by Huang et al.
(2016), R${}_{0}=8.3$ kpc and V${}_{0}=240$ km s-1. The agreement is very
good.

We commented before in sections 3.2.2 and 3.3 that there may be a positive
bias in the derived stellar surface gravities. The amount of bias may very
well be in the region of $\Delta\log{g}=+0.15$, as suggested by figure 6. The
remeasurements of just 7 objects find in favour of a larger bias (see section
3.3), but we do not otherwise see evidence that the distance range of our
stellar sample, strongly influenced by $\log{g}$, is significantly
underestimated (cf. figure 9 and the linked discussion). We have also
considered whether there is a bias in the measured RV. The best evidence we
have of this comes from the 7 remeasurements at red and blue wavelengths: the
weighted mean offset obtained is $-2.7$ km s-1. If all stellar surface
gravities are corrected down by 0.15 dex, and this potential RV bias is also
taken out uniformly, the trend in circular speed aquires the form of the blue
line in figure 14. The main impact of these changes is to stretch the results
out to an increased Galactocentric radius, in response to the $\log{g}$ shift.
The outcome remains consistent with the Huang et al. (2016) results. Another
factor playing a role is the presumed metallicity for the A stars: the orange
line in the figure shows the rotation curve when setting $[$Fe/H$]=-0.5$ for
the A stars. It can be seen that this change also has little impact.

So we have that both the clump giants and, now, our A/F star sample favour a
rotation law that rises out to $R_{G}\sim 14$ kpc, after a minimum near $\sim
11$ kpc. But we have also demonstrated how distance error can modify the
observed RV-distance trend. And indeed Binney & Dehnen (1997) presented a
thought experiment that drew attention to how a linear increase in Galactic
rotation into the outer disc would arise this way. The particular example they
presented was of the inferred law from tracers confined within a ring –
mimicking gas tracers associated with spiral arms – at 1.6 times the Sun’s
Galactocentric radius. These were subject to distance uncertainties similar to
the larger errors considered in the lower panel of figure 12. This extreme is
avoided here. The facts of the spread in stellar parameters ($\Delta M_{i}\sim
2$) and apparent magnitudes ($\Delta m_{i}\sim 1.5$, see figure 3) combine
with gently rising extinction (see figure 9) to yield an underlying stellar
distribution that should span at least 5 kpc – nor is there an expectation
these stars would be confined to e.g. just the Perseus Arm. This leaves us
cautiously supporting the case for an increase in circular speed outside the
Solar Circle and looking forward to the more extensive studies needed to
clarify the situation.

Figure 14:  Galactic disc circular speed results from Huang et al. (2016) are
reproduced in red. The circular speeds derived from the mean RV trend for
results at $\ell=118^{\circ}$, adopting solar metallicity, are in green.
Results obtained with $\Delta\log{g}=-0.15$ and $\Delta$RV$=-2.7$ km s-1
potential bias corrections are shown in blue. Orange is used for the results
obtained when the A star parameters for $[$Fe/H$]=-0.5$ are used in place of
solar metallicity parameters. The shaded region around each HectoSpec line
represents its error – propagated from the error on the mean trend in figure
11.

###  5.3 Spiral arm perturbations

Spiral arms in galactic discs are widely viewed as linked with non-
axisymmetric kinematic perturbations. To assess whether our data can expose
such an effect, we examine the $\ell=178^{\circ}$ sight line, since it is
close to the radial direction that minimises shear due to Galactic rotation
and more easily reveals low-amplitude perturbations that may be associated
with spiral arm structure. Monguió, Grosbøl & Figueras (2015) used B4-A1 stars
to find a stellar overdensity due to the Perseus spiral arm at a heliocentric
distance of $1.6\pm 0.2$ kpc in the anticentre direction, and Reid et al.
(2014) used parallaxes of 24 star forming regions to find the arm to be
located at $2$ kpc. The Perseus Arm is therefore located just short of the
sampled region in this work. However Reid et al. (2014) also found evidence
that an Outer Arm is located at a heliocentric distance of $6$ kpc in the
anticentre direction. This arm lies near the far end of our sampled region.

The scale of radial velocity perturbation depends on the model adopted for the
origin of the perturbation. On the one hand, Monari, Famaey & Siebert (2016)
simulated the effect of a spiral potential on an axisymmetric equilibrium
distribution function (emulating the Milky Way thin stellar disc) and found
radial velocity perturbations of order $-5$ km s-1 within the arms and $+5$ km
s-1 in between them. On the other, Grand et al. (2016) favour the transient
winding arm view and find the perturbation of young stars ($<3$ Gyr) to be
considerably stronger at up to $20$ km s-1.

We do not see any clear signs of perturbations, negative or positive, alligned
with the Outer Arm in the HectoSpec results. However since the HectoSpec RV
error is comparable with predicted perturbations at the lower end of the
expected range, and the stellar sample is subject to distance error, it is not
obvious that we should expect to observe their signal in our data. In order to
test this, we conducted another simulation. A sample of stars the same size as
the HectoSpec $\ell=178^{\circ}$ sample and with the same distance
distribution were assigned RVs according to a sinusoidal waveform – three
separate tests were conducted with amplitudes of 5, 10 and 20 km s-1. Two
spiral arms were put at 2 kpc and 6 kpc. Velocity scatter, RV error and
distance error were then applied to the distribution, and a mean trend was
computed as in section 4.1.

The results obtained for the small perturbations (5 km s-1) exhibited no clear
signs of the input sinusoid’s phasing or wavelength, giving a mean trend
compatible with zero for most of the sampled range. In contrast, the results
for the larger amplitude perturbations did show signs of the input phasing,
wavelength and amplitude. We conclude that spiral arm perturbations of small
amplitude would be unlikely to appear with clear statistical significance in
our results, but those of larger amplitude would.

Returning to the HectoSpec $\ell=178^{\circ}$ results themselves (figure 10),
we do observe a wave-like structure in the mean RV trend with amplitude $\sim
5-10$ km s-1, but not at an implied phase or long-enough wavelength that would
make sense in comparison with the expected locations of the Perseus Arm and
Outer arms (at $\sim 2$ kpc and $\sim 6$ kpc in this sightline, respectively).
If the RV wobble is real, rather than a sample size effect, it is most likely
a local effect unconnected with the larger scale structure of the Galactic
disc. But we must discard the possibility of large amplitude (10-20 km s-1)
perturbations in this direction. The results for this sightline are in keeping
with the findings of Fernández, Figueras & Torra (2001), who used both OB
stars and Cepheids to limit perturbations to under 3 km s-1.

Finally we comment on the form of the $\ell=118^{\circ}$ results (figure 11).
Specifically, can a spiral arm perturbation explain the observed deviation
from the trend predicted by a flat rotation law? To make this comparison, the
same three simulations as described above for the $\ell=178^{\circ}$ sightline
were conducted for $\ell=118^{\circ}$. In this case the spiral arms are
located at $3$ and $6.5$ kpc (Reid et al., 2014) and account is taken of the
expected combination of Galactic azimuthal and radial perturbations along the
line of sight. In this case the observed RV trend, when compared with the
simulation alternatives, shows a deviation from the flat rotation law of a
scale similar to that of the largest ($20$ km s-1) perturbation investigated.
This stands in clear contrast to the low-amplitude perturbation compatible
with the $\ell=178^{\circ}$ sightline. Hence it appears there is no simple
coherent way of interpreting our radial velocity trends in terms of one class
of spiral arm perturbation model.

##  6 Conclusions

For the first time, we have demonstrated that A/F star spectroscopy, even when
restricted to a small region around the Calcium triplet lines, can provide
useful insights into the kinematics of the Milky Way disc. With the use of the
method employed in this paper, stellar parameters and radial velocities can be
measured for the very large samples of these stars, with $i<18$ (Vega), that
will be accessible to future massive-multiplex instruments such as the WEAVE
and 4MOST spectrographs being constructed for the William Herschel and VISTA
telescopes respectively.

This first result, deploying a dense sample of $\sim$800 A/F stars within just
one pencil beam, 1 degree in diameter, at $\ell=118^{\circ}$, fits in well
with the trend of a rising mean circular speed within the Galactic disc
reported recently by Huang et al. (2016), over the Galactocentric radius range
$11<$ RG (kpc) $<15$. Interpreting these RV measurements in terms of spiral
arm perturbation is more difficult as then it becomes hard to reconcile the
$\ell=118^{\circ}$ and $\ell=178^{\circ}$ results. The Huang et al. (2016)
sample of clump giants is much larger, comprising $\sim$16000 stars, drawn
from a very broad fan of outer-disc longitudes
($100^{\circ}<\ell<230^{\circ}$, roughly) – hence their results describe a
longitude average, in contrast to ours. Further dense-sampling studies like
ours will be needed to discover whether the rotation law is closely
axisymmetric as has usually been assumed, hitherto. This first comparison
passes the test. However, both stellar results are at variance with the maser-
based study of Reid et al. (2014) which obtained evidence of a very nearly
flat rotation law to $R_{G}=16$ kpc, as well as with the long-established
Brand & Blitz (1993) law. The origin of this difference may rest in the very
limited maser data at all longitudes beyond R${}_{G}=12$ kpc (see figure 1 of
Reid et al., 2014). On the other hand, caution still needs to be exercised
that some of the inferred steepening may prove to be a consequence of
uncertainties in stellar distances derived from spectroscopic parallax. If the
steepening is correct, it would lend support to previous evidence of a ring of
dark matter in the outer disc at RG between 13 and 18 kpc (e.g see Kalberla,
Dedes & Kerp, 2007).

Our goal is to open up the use of A/F stars in studies of Galactic disc
structure at red wavelengths that mitigate the effects of extinction. It is
more common to derive stellar parameters for these earlier type stars in the
blue optical and so it was not evident at the outset whether fitting to the
red range only would suffer from biases or degeneracies in parameter
determination. For the present purpose, it has become clear that degeneracies
are less troublesome for A stars than F stars, although it would be desirable
in future to better disentangle effective temperature and metallicity, even
for A stars – better spectral resolution and wider wavelength coverage can
both help here. We plan to examine these options. The basic method of target
selection, using IPHAS colours, is certainly very efficient (see also Hales et
al., 2009). We have assessed possible bias in our numerical results: whilst
there appears to be some, our tests examining their impact have so far shown
that the impact is modest. Furthermore, we find that the random error on the
mean trend in circular speed with increasing Galactocentric radius, achieved
with our dense localised sampling, is generally better than $\pm 5$ km s-1,
and represents an advance on previous measurements.

Since the single biggest source of uncertainty in a study like this originates
in the stellar distances, we look forward to the new opportunity that starts
to take shape with the stellar astrometry in the Gaia DR2 release in April
2018. A representative $i$ magnitude among our selected A/F stars is $\sim 17$
(or Gaia $G$ approaching 18): at this brightness, the anticipated DR2 (end-of-
mission) parallax and proper motion errors are likely to be 150 (100) $\mu$as,
and 80 (50) $\mu$as yr-1, respectively (see ESA website and Katz & Brown,
2017). Whilst distances can not be nailed down for individual stars, over the
range of interest here from $\sim$2 to almost 10 kpc (parallaxes of 500 down
to 100 $\mu$as), significantly improved constraints will become available.
There will also be the opening to bring into account full space motions, on
folding in Gaia proper motion data. As our next step to prepare for this, we
are building customised forward simulations based on Galactic models aimed at
predicting full space motion distributions. As part of this we will examine in
further detail the roles played by our methods of target selection and
analysis. We have already checked one of the more obviously significant issues
that concerns us - no weeding of binary stars - and have found that to be a
small effect.

Another experiment for the future will be to examine the differences in
performance of A-star and clump-giant samples, on the same sightline. Based on
what we have learned here, there is reason to expect A-star samples to be
especially valuable in characterising the outer Galactic disc, where these
younger less-scattered stars are relatively more common.

## Acknowledgments

AH and HF acknowledge support from the UK’s Science and Technology Facilities
Council (STFC), grant nos. ST/K502029/1 and ST/I505699/1. JED and MM
acknowledge funding via STFC grants ST/J001333/1 and ST/M001008/1. JJD was
funded by NASA contract NAS8-03060 to the Chandra X-ray Center. SES
acknowledges support via STFC grant no. ST/M00127X/1. NJW acknowledges an STFC
Ernest Rutherford Fellowship, grant no. ST/M005569/1.

Observations reported here were obtained at the MMT Observatory, a joint
facility of the Smithsonian Institution and the University of Arizona. This
study has also benefitted from the service programme (ref. SW2016b05) on the
William Herschel Telescope (WHT): the WHT and its service programme are
operated on the island of La Palma by the Isaac Newton Group of Telescopes in
the Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias.

We thank the referee of this paper for useful comments that have helped us
improve the paper’s content.

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${P}$${Q_{\overline{a}}}$${\mathscr{S}_{m}}$${\mathscr{S}_{n}}$$\scriptstyle{w}$$\scriptstyle{m_{\star}}$$\scriptstyle{n_{\star}}$$\scriptstyle{\psi_{q}}$$\scriptstyle{f}$${LP}$${\overline{a}}$${L\mathscr{S}_{n}}$$\scriptstyle{\vartheta}$$\scriptstyle{Ln_{\star}}$$\scriptstyle{L(f\circ\psi_{q})}$

Hence, precomposing with $\mu_{2}$ we get

$Ln_{\star}\circ\mu_{2}=L(f\circ\psi_{q})\circ\vartheta\circ\mu_{2}=L(f\circ\psi_{q})\circ\mu_{1}.\qed$

For convenience of notation, let us write

$\widetilde{\gamma}_{q}\coloneqq L(f\circ\psi_{q})\colon\overline{a}\to
L\mathscr{S}_{n}\ \text{ and }\ \gamma_{q}\coloneqq\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\widetilde{\gamma}_{q}\colon\overline{a}\to e.$

With this notation we have

$\displaystyle\gamma_{q}\circ\mu_{1}$ $\displaystyle=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\widetilde{\gamma}_{q}\circ\mu_{1}$
$\displaystyle=\varepsilon_{e}\circ L\mathfrak{i}_{n}\circ
Ln_{\star}\circ\mu_{2}$ Lemma 6.8 $\displaystyle=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\tau_{2}\circ\nu$

and so the leftmost diagram below commutes.

${\overline{LP}}$${\overline{L\mathscr{S}_{n}}}$${\overline{a}}$${e}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\tau_{2}}$$\scriptstyle{\gamma_{q}}$${\overline{LP}}$${\overline{L\mathscr{S}_{n}}}$${\overline{a}}$${LQ^{\prime}}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\sigma_{1}}$${\ulcorner}$

Now, note that by (E2) the top face of diagram (3), displayed in the rightmost
diagram above, is a pushout in $\operatorname{\mathscr{E}}$. By the universal
property of pushouts, there is a unique $\delta_{q}\colon LQ^{\prime}\to e$
satisfying

$\delta_{q}\circ\sigma_{1}=\gamma_{q}\ \text{ and }\
\delta_{q}\circ\tau_{1}=\varepsilon_{e}\circ L\mathfrak{i}_{n}\circ\tau_{2}.$

Define $\zeta_{q}\coloneqq(\delta_{q})^{\flat}\colon Q^{\prime}\to Re$ for all
path embeddings $q\colon Q^{\prime}\rightarrowtail R(a+_{LP}L\mathscr{S}_{n})$
above $\xi$. Further, if $q$ is below $\xi$, we let $\zeta_{q}$ be the obvious
restriction of $\zeta_{\xi}$.

######  Lemma 6.9.

The following family of arrows forms a compatible cocone over the diagram of
path embeddings into $\mathscr{S}_{\xi}$:

$\\{\zeta_{q}\mid q\colon Q^{\prime}\rightarrowtail
R(a+_{LP}L\mathscr{S}_{n})\ \text{is a path embedding comparable with
$\xi$}\\}.$

###### Proof.

Fix arbitrary path embeddings

$q\colon Q^{\prime}\rightarrowtail R(a+_{LP}L\mathscr{S}_{n})\ \text{ and }\
q^{\prime}\colon Q^{\prime\prime}\rightarrowtail R(a+_{LP}L\mathscr{S}_{n})$

comparable with $\xi$. The compatibility condition for the cocone states that
$\zeta_{q}$ extends $\zeta_{q^{\prime}}$ whenever $q\geq q^{\prime}$. It
suffices to settle the case where $\xi\leq q\leq q^{\prime}$. Also, it is
enough to show that $\delta_{q}$ extends $\delta_{q^{\prime}}$, i.e.
$\delta_{q}\circ L{!}=\delta_{q^{\prime}}$ where ${!}\colon
Q^{\prime\prime}\rightarrowtail Q^{\prime}$ is the unique embedding. Just
observe that $\delta_{q}\circ L{!}=\delta_{q^{\prime}}$ entails

$\zeta_{q}\circ{!}=\delta_{q}^{\flat}\circ{!}=((\delta_{q}^{\flat}\circ{!})^{\\#})^{\flat}=(\delta_{q}\circ
L{!})^{\flat}=\delta_{q^{\prime}}^{\flat}=\zeta_{q^{\prime}}.$

Consider the following diagram all whose vertical faces are pullbacks and note
that by (A4), and more precisely by item (ii) in Definition 5.8, the pullback
of $L!$ along $\sigma_{1}$ is of the form $L\ell$ for some arrow $\ell\colon
Q_{\overline{\overline{a}}}\to Q_{\overline{a}}$.

${\overline{\overline{LP}}}$${\overline{\overline{L\mathscr{S}_{n}}}}$${\overline{\overline{a}}}$${LQ^{\prime\prime}}$${\overline{LP}}$${\overline{L\mathscr{S}_{n}}}$${\overline{a}}$${LQ^{\prime}}$${LP}$${L\mathscr{S}_{n}}$${e}$${a}$${a+_{LP}L\mathscr{S}_{n}}$$\scriptstyle{\overline{\tau}_{2}}$$\scriptstyle{\overline{\tau}_{1}}$$\scriptstyle{\overline{\sigma}_{1}}$$\scriptstyle{L{!}}$$\scriptstyle{\delta_{q^{\prime}}}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{L\ell}$$\scriptstyle{\delta_{q}}$$\scriptstyle{Ln_{\star}}$$\scriptstyle{m^{\\#}}$$\scriptstyle{\lambda}$$\scriptstyle{\iota}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{q^{\\#}}$

In view of the definition of $\delta_{q^{\prime}}$ in terms of the universal
property of pushouts, it suffices to show that $\delta_{q}\circ L{!}$
satisfies

$(\delta_{q}\circ L{!})\circ\overline{\sigma}_{1}=\gamma_{q^{\prime}}\ \text{
and }\ (\delta_{q}\circ L{!})\circ\overline{\tau}_{1}=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\tau_{2}\circ\overline{\tau}_{2}.$

The latter equation follows at once from the identity
$\delta_{q}\circ\tau_{1}=\varepsilon_{e}\circ L\mathfrak{i}_{n}\circ\tau_{2}$.
With regards to the former, we have

$(\delta_{q}\circ
L{!})\circ\overline{\sigma}_{1}=\delta_{q}\circ\sigma_{1}\circ
L\ell=\gamma_{q}\circ L\ell.$

Thus it suffices to show that $\gamma_{q}\circ L\ell=\gamma_{q^{\prime}}$, and
this clearly follows if we prove that $\widetilde{\gamma}_{q}\circ
L\ell=\widetilde{\gamma}_{q^{\prime}}$. Recall that $\psi_{q^{\prime}}$ is the
unique morphism such that the composite

${Q_{\overline{\overline{a}}}}$${\mathscr{S}_{m}}$${Ra}$$\scriptstyle{\psi_{q^{\prime}}}$$\scriptstyle{\mathfrak{i}_{m}}$

coincides with $(\sigma_{2}\circ L\ell)^{\flat}$. But

$(\sigma_{2}\circ
L\ell)^{\flat}=((\sigma_{2}^{\flat}\circ\ell)^{\\#})^{\flat}=\sigma_{2}^{\flat}\circ\ell,$

so $\mathfrak{i}_{m}\circ\psi_{q}\circ\ell=\sigma_{2}^{\flat}\circ\ell$
entails $\psi_{q^{\prime}}=\psi_{q}\circ\ell$. Therefore,

$\widetilde{\gamma}_{q^{\prime}}=L(f\circ\psi_{q^{\prime}})=L(f\circ\psi_{q}\circ\ell)=\widetilde{\gamma}_{q}\circ
L\ell.\qed$

The previous lemma entails the existence of a unique morphism
$h\colon\mathscr{S}_{\xi}\to Re$ satisfying $h\circ q=\zeta_{q}$ for all path
embeddings $q$ into $R(a+_{LP}L\mathscr{S}_{n})$ that are comparable with
$\xi$. In order to conclude that $\xi_{\star}\to n^{\prime}_{\star}$ as
desired, we prove the following useful property of the cocone consisting of
the morphisms $\zeta_{q}$.

######  Lemma 6.10.

Let $G\coloneqq LR$ and consider the composite morphism

${RL\mathscr{S}_{n}}$${RGe}$${Re.}$$\scriptstyle{RL\mathfrak{i}_{n}}$$\scriptstyle{R\varepsilon_{e}}$

If $q=R\lambda\circ\alpha$ for some arrow $\alpha\colon
Q^{\prime}\rightarrowtail RL\mathscr{S}_{n}$, then
$\zeta_{q}=R\varepsilon_{e}\circ RL\mathfrak{i}_{n}\circ\alpha$.

###### Proof.

Suppose that $q=R\lambda\circ\alpha$ for some $\alpha\colon
Q^{\prime}\rightarrowtail RL\mathscr{S}_{n}$. We have

$(R\varepsilon_{e}\circ
RL\mathfrak{i}_{n}\circ\alpha)^{\\#}=((\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\alpha^{\\#})^{\flat})^{\\#}=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\alpha^{\\#}.$

By the universal property of $\delta_{q}=\zeta_{q}^{\\#}$,
$\zeta_{q}=R\varepsilon_{e}\circ RL\mathfrak{i}_{n}\circ\alpha$ if, and only
if,

$(\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\alpha^{\\#})\circ\tau_{1}=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\tau_{2}$ (4)

and

$(\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\alpha^{\\#})\circ\sigma_{1}=\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ\widetilde{\gamma}_{q}.$ (5)

Observe that $\alpha^{\\#}\circ\tau_{1}=\tau_{2}$ because

$\lambda\circ\alpha^{\\#}\circ\tau_{1}=q^{\\#}\circ\tau_{1}=\lambda\circ\tau_{2}$

and $\lambda$ is a monomorphism. Thus, equation (4) holds. Further, note that

$q^{\\#}=(R\lambda\circ\alpha)^{\\#}=((\lambda\circ\alpha^{\\#})^{\flat})^{\\#}=\lambda\circ\alpha^{\\#}$

and so $\tau_{1}$ in diagram (3) is an isomorphism. As pushout squares of
embeddings in $\operatorname{\mathscr{E}}$ are also pullbacks by (E2),
$\mu_{1}$ in diagram (3) is also an isomorphism. Therefore,

$\displaystyle\lambda\circ\alpha^{\\#}\circ\sigma_{1}$
$\displaystyle=q^{\\#}\circ\sigma_{1}=q^{\\#}\circ\sigma_{1}\circ\mu_{1}\circ\mu_{1}^{-1}=\lambda\circ
Ln_{\star}\circ\mu_{2}\circ\mu_{1}^{-1}$
$\displaystyle=\lambda\circ\widetilde{\gamma}_{q}\circ\mu_{1}\circ\mu_{1}^{-1}=\lambda\circ\widetilde{\gamma}_{q}$

and so $\alpha^{\\#}\circ\sigma_{1}=\widetilde{\gamma}_{q}$. Equation (5) then
follows at once. ∎

Since $\xi=R\lambda\circ(Ln^{\prime}_{\star})^{\flat}$, recalling that we
identify $n^{\prime}_{\star}$ with a path embedding into $\mathscr{S}_{n}$, an
application of the previous lemma with
$\alpha\coloneqq(Ln^{\prime}_{\star})^{\flat}$ yields

$\zeta_{\xi}=R\varepsilon_{e}\circ
RL\mathfrak{i}_{n}\circ(Ln^{\prime}_{\star})^{\flat}=(\varepsilon_{e}\circ
L\mathfrak{i}_{n}\circ Ln^{\prime}_{\star})^{\flat}=(\varepsilon_{e}\circ
Ln^{\prime})^{\flat}=((n^{\prime})^{\\#})^{\flat}=n^{\prime}.$

In other words, $h\circ\xi_{\star}=n^{\prime}$ and so $\xi_{\star}\to
n^{\prime}$. It follows from Remark 4.16 that $\xi_{\star}\to
n^{\prime}_{\star}$, thus concluding the proof of Proposition 6.4.

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