# An Investigation of the Absolute Proper Motions of the SCUSS Catalog

Xiyan Peng11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012,
China 22affiliation: University of Chinese Academy of Sciences, Beijing,
100039, China , Zhaoxiang Qi33affiliation: Shanghai Astronomical Observatory,
Chinese Academy of Sciences, Shanghai 200030, China  , Zhenyu Wu11affiliation:
Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China ⋆⋆affiliationmark:  , Jun
Ma11affiliation: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, China , Cuihua
Du22affiliation: University of Chinese Academy of Sciences, Beijing, 100039,
China , Xu Zhou11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012,
China , Yong Yu33affiliation: Shanghai Astronomical Observatory, Chinese
Academy of Sciences, Shanghai 200030, China  , Zhenghong Tang33affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
200030, China  , Zhaoji Jiang11affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Hu Zou11affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Zhou Fan11affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Xiaohui Fan44affiliation: Department of Astronomy and
Steward Observatory, University of Arizona, Tucson, Arizona, USA , Martin C.
Smith33affiliation: Shanghai Astronomical Observatory, Chinese Academy of
Sciences, Shanghai 200030, China  , Linhua Jiang55affiliation: Kavli Institute
for Astronomy and Astrophysics, Peking University, Beijing 100871, China ,
Yipeng Jing66affiliation: Department of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240 , Mario G. Lattanzi77affiliation:
INAF–Osservatorio Astrofisico di Torino, Strada Osservatorio 20, 10025 Pino
Torinese, TO, Italy. , Brian J. McLean88affiliation: Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA. , Michael
Lesser44affiliation: Department of Astronomy and Steward Observatory,
University of Arizona, Tucson, Arizona, USA , Jundan Nie11affiliation: Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, China , Shiyin Shen33affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
200030, China  , Jiali Wang11affiliation: Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100012, China , Tianmeng Zhang11affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Zhimin Zhou11affiliation: Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing, 100012, China , Songhu Wang99affiliation: School of Astronomy and
Space Science and Key Laboratory of Modern Astronomy and Astrophysics in
Ministry of Education, Nanjing University, Nanjing 210093<EMAIL_ADDRESS>
###### Abstract

Absolute proper motions for $\sim$ 7.7 million objects were derived based on
data from the South Galactic Cap $u$-band Sky Survey (SCUSS) and astrometric
data derived from uncompressed Digitized Sky Surveys that the Space Telescope
Science Institute (STScI) created from the Palomar and UK Schmidt survey
plates. We put a great deal of effort into correcting the position-,
magnitude- and color- dependent systematic errors in the derived absolute
proper motions. The spectroscopically confirmed quasars were used to test the
internal systematic and random error of the proper motions. The systematic
errors of the overall proper motions in the SCUSS catalog are estimated as
$-0.08$ and $-0.06\,\rm{mas/yr}$ for $\mu_{\alpha}\rm{cos}\delta$ and
$\mu_{\delta}$, respectively. The random errors of the proper motions in the
SCUSS catalog are estimated independently as 4.2 and $4.4\,\rm{mas/yr}$ for
$\mu_{\alpha}\rm{cos}\delta$ and $\mu_{\delta}$. There is no obvious
position-, magnitude- and color- dependent systematic errors of the SCUSS
proper motions. The random error of the proper motions goes up with the
magnitude from about $3\,\rm{mas/yr}$ at $u=18.0\,\rm{mag}$ to about
$7\,\rm{mas/yr}$ at $u=22.0\,\rm{mag}$. The proper motions of stars in SCUSS
catalog are compared with those in the SDSS catalog, and they are consistent
well.

astrometry–catalog–stars:kinematics

##  1 INTRODUCTION

A catalog of proper motions in the International Celestial Reference System
(ICRS), with precise positions and proper motions as well as deep limiting
magnitude, is important for research on the Galactic structure, Galactic
kinematics and stellar populations.

There are two methods to determine the absolute proper motions of stars based
on the hypothesis that the proper motions of background galaxies are zero. One
is that all the observations obtained in different epochs are transformed into
one reference position system using galaxies, and absolute proper motions are
calculated directly. For example, Munn et al. (2004) took this method to
construct the improved SDSS proper motion catalog combining USNO-B (Monet et
al. 2003) and SDSS (Gunn et al. 1998; York et al. 2000; Lupton et al. 2001;
Stoughton et al. 2002). The basic procedure in this work is as follows. For
each object from USNO-B, the nearest 100 galaxies (classified as $galaxy$ by
SDSS morphological
classification111http://www.sdss.org/dr1/products/catalogs/flags.html)
detected in both SDSS and USNO-B were selected. Then the mean offsets in right
ascension and declination between the SDSS and USNO-B positions were measured
from the 100 nearest galaxies, and were added to the USNO-B position of this
object. Finally the proper motions of objects in SDSS were computed using a
linear fit with the re-calibrated USNO-B positions and the SDSS positions at
different epochs.

The other method to determine absolute proper motions is to take a novel
calibration method which iteratively uses the stellar sources and galaxies on
each position system to eliminate all systematic errors between different
position systems. Absolute Proper motions Out side the Plane (APOP) adopted
this technique with the Guide Star Catalog II (GSC-II) Schmidt plate data (Qi
et al. 2012; Lasker et al. 2008). Meanwhile, the APOP used the measured
coordinates ($x$, $y$) of objects instead of equatorial coordinates to
calculate the absolute proper motions. The principal calibration steps for
APOP are as follows.

  1. 1.

Removing the position-dependent systematic errors (PdE) with a moving-mean
filter using good-quality stellar objects;

  2. 2.

Selecting galaxies from non-point-like sources (non-stars) via their common
null-motion characteristics;

  3. 3.

Calibrating magnitude- and color- dependent (MdE and CdE) errors and the
residual PdE of all objects with respect to the galaxies selected above;

  4. 4.

Calculating absolute proper motions from all-epoch plate data with a linear
fit.

The construction of the latest version of APOP catalog and evaluation of the
catalog are described in detail by Qi et al. (2014). In this paper, we take
almost the same method and pipeline as APOP to derive absolute proper motions.

The SCUSS (Zou et al. 2014; Zhou et al. 2014; Jia et al. 2014) is a $u$-band
(3538 Å) imaging survey using the $2.3\,\rm{m}$ Bok telescope. The survey
covers about 5000 square degrees. Most of the SCUSS fields were observed twice
in the course of survey from 2010 to 2013. It is difficult to measure accurate
proper motions for the SCUSS objects from the SCUSS data only. In this paper,
we construct a SCUSS proper motion catalog by combining the SCUSS data with
the GSC-II Schmidt plate data. The GSC-II Schmidt plate data is an all-sky
astro-photometric data base which were derived from the digitization of the
Palomar and UK Schmidt surveys. The GSC-II contains astrometry, photometry and
classification for objects down to the limiting magnitude of the plates .

This paper is organized as follows. In Section 2, we summarize the SCUSS
program and GSC-II project that are used to derive the SCUSS proper motion
catalog. Section 3 describes the construction of the SCUSS proper motion
catalog. In Section 4, quasars are used to perform internal test of the proper
motions. We compare the catalog with the SDSS proper motion catalog in Section
5. We summarize the paper in Section 6.

##  2 DATA

###  2.1 The SCUSS Program

The SCUSS is a $u$-band imaging survey with an effective wavelength of $\sim$
3538 Å and a bandwidth $\sim$ 345 Å . This $u$-band filter is slightly bluer
than the SDSS $u$-band filter. The imaging depth is $\sim$ $23.5\,\rm{mag}$,
roughly $1.5\,\rm{mag}$ deeper than that of the SDSS imaging survey. The SCUSS
project began its observations in September 2010 and ended in October 2013. It
covers a total 5000 square degrees. Proper motions are measured with respect
to a reference frame of galaxies. Hence, the accuracy of star/galaxy
classification affects systematic error of the proper motions. The SCUSS
proper motion catalog covers 3700 square degrees (Galactic latitude
$b<-30^{\circ}$ and Equatorial latitude $\delta>-10^{\circ}$). In this area,
the accuracy of star/galaxy classification is reliable. The spatial
distribution of the SCUSS proper motion catalog is shown in Figure 1. Most of
the images in this area were taken with seeing $<3^{\prime\prime}$, sky
background $<500$ ADU.

Figure 1: Spatial distribution of the SCUSS proper motion catalog in a J2000
equatorial reference frame. The SCUSS proper motion catalog covers 3700 square
degrees (Galactic latitude $b<-30^{\circ}$ and Equatorial latitude
$\delta<-10^{\circ}$).

The description of reduction of the SCUSS data can be found in Zou et
al.(2014). Photometry was done using SExtractor (Bertin & Arnouts 1996). The
measured $x$ and $y$ positions were calculated by applying SExtractor to
sources brighter than 1$\sigma$ of the background level. External astrometric
errors were determined by crossing-matching the SCUSS sources to the UCAC4
sources (Zacharias et al. 2013). Over the whole survey, the mean offsets of
right ascension and declination between the SCUSS and UCAC4 are
$\triangle\alpha=-0^{\prime\prime}.001\pm 0^{\prime\prime}.008$,
$\triangle\delta=-0.^{\prime\prime}001\pm 0^{\prime\prime}.007$, respectively.
In this work, we will construct a proper motion catalog using the measured
coordinates ($x$, $y$), the celestial coordinates ($\alpha$, $\delta$), the
$u$-band magnitudes from SCUSS and the star/galaxy morphological
classifications from SDSS for objects on SCUSS images. The basic information
for each frame including plate size and scale as well as observation time is
also needed in the process of calculation of proper motions.

###  2.2 Astronomic Data from GSC-II

GSC-II (Lasker et al. 2008), as first epoch data to obtain the proper motions
of SCUSS, was originally taken from the survey plates observed by the Palomar
Schmidt telescope and the UK Schmidt telescope. Each individual plate was
digitized by the Guide Star Automatic Measuring Machine (GAMMA) and was
processed through image processing and plate calibration. The final products
were stored in the COMPASS database (Lasker et al. 2008). GSC-II contains
astrometry, multi-band photometry ($R_{F}$, $B_{J}$, $I_{N}$) and star/non-
star classification for objects down to the limiting magnitude of the survey
plates. For stellar sources, the all-sky average absolute position error per
coordinate ranges from $0.^{\prime\prime}2$ to $0.^{\prime\prime}28$ depending
on magnitude. Stellar photometry is determined to $0.13-0.22\,\rm{mag}$ as a
function of magnitude and photographic passbands ($R_{F}$, $B_{J}$, $I_{N}$).
Outside of the Galactic plane, the catalogue is complete to
$R_{F}=20\,\rm{mag}$. There are about 1210 Schmidt plates from different
surveys used in this paper. The main characteristics of the surveys used in
this paper including the epoch, band, limiting magnitude and sky coverage as
well as the number of plates used in this work are summarized in Table 1. The
basic information for objects on each plate including the measured
coordinates, the equatorial coordinates, magnitude and classification
(star/non-star) is needed in this paper. In addition, The basic information
for each plate from GSC-II data base including telescope name, plate size and
scale, observation time is also needed.

##  3 CONSTRUCTION OF THE SCUSS PROPER MOTION CATALOG

Two major aspects determine the final accuracy of absolute proper motions. One
is the measured uncertainties of object positions on the images of all epochs,
and the other one is the systematic error between different images. The latter
one requires unifying images in different epochs and different bands to one
reference system (Qi et al. 2012).

The general steps of the construction of the SCUSS proper motion catalog are
similar to those of APOP. Relevant equations and details are recommend to see
the paper of Qi et al. (2014). A few steps are slightly different. The steps
are summarized as follows.

  1. 1.

Extract all the GSC-II plates and SCUSS images in the same sky field;

  2. 2.

Compare GSC-II reference plates with the SDSS, update the star/non-star
classification in GSC with star/galaxy in the SDSS;

  3. 3.

Reject the synthetic double stars (stars with the same ID) in each GSC-II
plate;

  4. 4.

Find common objects between the program images and the reference plate;

  5. 5.

Smooth out position-dependent difference between each image pair using stars
with small measuring error;

  6. 6.

Regenerate the $x$, $y$ of all the objects on program plates using galaxies
and remove the MdE, CdE and residual PdE of proper motions;

  7. 7.

Calculate the absolute proper motions and new positions in a given epoch using
all the images.

The plates, which are from GSC-II with good quality and overlap with the
SCUSS, were chosen as reference plates. The star/galaxy classifications of
sources play an important role in the calculation of the proper motions.
However, in GSC-II, the probability of stars correctly classified drops with
magnitude from better than $90\%$ at $R_{F}=18.0\,\rm{mag}$ to less than
$50\%$ at $R_{F}=20.0\,\rm{mag}$, while the percentage that a galaxy is
classified as non-star remains above $90\%$ for all the magnitude range.
Outside of the Galactic plane, stellar classification is reliable to at least
90% confidence for magnitudes brighter than $R_{F}=19.5\,\rm{mag}$ (Lasker et
al. 2008). The low accuracy of star/non-star classification of the fainter
objects of the GSC-II reference plates eventually affect the error of the
proper motions. In our work, most of the SCUSS survey are covered by the SDSS,
star/galaxy separation of SDSS is about $95\%$ correct at 21.0 mag in $r$ band
(Stoughton et al. 2002). So, we update the GSC-II star/non-star classification
with the star/galaxy classification from SDSS morphological classification.

##  4 PROPERTIES OF SCUSS PROPER MOTION CATALOG

The SCUSS proper motion catalog contains the equatorial coordinates ($\alpha$,
$\delta$), the absolute proper motions ($\mu_{\alpha}\cos\delta$,
$\mu_{\delta}$), and the SCUSS $u$-band magnitudes for about 7.7 million
objects. The proper motions of quasars cannot be detected by present
astrometric observation in the optical band, so the intrinsic proper motions
of quasars can be considered as zero (Wu et al. 2011). Therefore, the measured
proper motions of quasars can be used to derive the systematic and random
error of the proper motion.

We choose the spectroscopically confirmed quasars from SDSS DR9 (Ahn et al.
2012). A total of 22831 common quasars among the SDSS and SCUSS catalog are
selected as a quasar sample. The gaussian function is used to fit the
distribution of the quasar proper motions to derive the systematic and random
error. The systematic error of proper motions is derived from the mean proper
motion of the quasar sample. The dispersion of the proper motions of quasars
can be considered as the random error of proper motions. Figure 2 displays the
histograms of the measured proper motions in right ascension
$\mu_{\alpha}\cos\delta$ (upper) and in declination $\mu_{\delta}$ (lower) for
all SDSS spectroscopically confirmed quasars in SCUSS proper motion catalog.
The random errors of the proper motions are 4.2 and $4.4\,\rm{mas/yr}$ in
$\mu_{\alpha}\cos\delta$ and $\mu_{\delta}$, respectively. The systematic
error of $\mu_{\alpha}\cos\delta$ is about $-0.08\,\rm{mas/yr}$ and the
systematic error of $\mu_{\delta}$ is about $-0.06\,\rm{mas/yr}$. Next, all
the 22831 quasars are used to test the position-, magnitude- and color-
dependent errors of the SCUSS proper motions.

Figure 2: The proper motion distributions of SDSS spectroscopically confirmed
quasars in SCUSS catalog. The upper panel is for $\mu_{\alpha}\cos\delta$ and
the lower panel is for $\mu_{\delta}$. Only quasars with the absolute values
of proper motions less than $50\,\rm{mas/yr}$ are used for the Gaussian
function fitting. The red solid line in each panel is the best-fitting
Gaussian function. The best-fitting parameters of the Gaussian function
including the means and the dispersions are labeled in each panel.

###  4.1 Influence of the Position on the Proper Motion

The scatters of $\mu_{\alpha}\cos\delta$ (upper) and $\mu_{\delta}$ (lower) of
the selected quasar sample as a function of $\alpha$ and $\delta$ are shown in
Figures 3 and 4, respectively. In order to derive the position-dependent error
of the proper motions, quasars are divided into different position groups with
a group size of $5^{\circ}$ (in declination) and $10^{\circ}$ (in right
ascension). In Figures 3 and 4, red points and red error bars show the
systematic and random errors of the quasar proper motions in different groups.
As mentioned above, the systematic and random errors correspond to the mean
and dispersion of the best fit Gaussian function of the distribution of proper
motions for quasar groups.

Figure 3 shows that, for $\mu_{\alpha}$cos$\delta$, there is no significant
dependence of the systematic or random errors in the $\alpha$ direction. The
systematic errors of $\mu_{\alpha}$cos$\delta$ for quasar groups are small and
fluctuate around zero. The random errors of $\mu_{\alpha}$cos$\delta$ are
about $4\,\rm{mas/yr}$. In addition, there is also no significant dependence
of the errors of $\mu_{\delta}$ in $\alpha$ direction. The systematic errors
of $\mu_{\delta}$ fluctuate around zero. The random errors of $\mu_{\delta}$
are around $4\,\rm{mas/yr}$ and do not significantly change with $\alpha$.
From Figure 4, we can see that there is no significant relation between error
(systematic and random error) of proper motions and $\delta$. The absolute
values of systematic errors of $\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$
for most quasar groups are less than $0.1\,\rm{mas/yr}$ and fluctuate around
zero. The random errors of proper motions are about $4\,\rm{mas/yr}$. So, from
Figures 3 and 4, we can find that the PdE of SCUSS proper motion catalog is
well eliminated and the random errors of the proper motions do not
significantly change with position.

Figure 3: The scatter of proper motions $\mu_{\alpha}$ cos $\delta$ (upper)
and $\mu_{\delta}$ (lower) for SDSS spectroscopically confirmed quasars in
SCUSS catalog as a function of coordinates in fields located in the right
ascensions from $0^{\circ}$ to $70^{\circ}$ and from $310^{\circ}$ to
$360^{\circ}$. The systematic and random errors are showed as red points and
red error bars in each panel.

Figure 4: The scatter of proper motions $\mu_{\alpha}$ cos $\delta$ (upper)
and $\mu_{\delta}$ (lower) for SDSS spectroscopically confirmed quasars in
SCUSS catalog as a function of coordinates in fields located in the
declination from $-10^{\circ}$ to $30^{\circ}$. The systematic and random
errors are showed as red points and red error bars.

###  4.2 Influence of the Magnitude and Color on the Proper Motion

Figure 5 shows the histogram of SCUSS $u$-band magnitude distribution of
quasar sample. The quasar sample covers the SCUSS $u$-band magnitude from 16.0
to $24.0\,\rm{mag}$. Quasars are divided into different magnitude groups with
a group size of 1 mag. Figure 6 shows the scatters of
$\mu_{\alpha}$cos$\delta$ (upper) and $\mu_{\delta}$ (lower) of quasar sample
as a function of SCUSS $u$-band magnitude. In Figure 6, the red points and red
error bars show the systematic and random errors of quasar proper motions in
different groups.

Figure 5: The SCUSS $u$-band magnitude distribution for SDSS spectroscopically
confirmed quasars in SCUSS catalog.

From Figure 6, we can see that no MdE of the proper motions is found. The
systematic errors of $\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$ are both
small. The random error of proper motions increases from the brightest end of
the magnitude group to the faintest end of the magnitude group. For
$\mu_{\alpha}$cos$\delta$, the random error goes up from $2.5\,\rm{mas/yr}$
for quasars brighter than $u=18\,\rm{mag}$ to $7.5\,\rm{mas/yr}$ for quasars
fainter than $u=22\,\rm{mag}$. For $\mu_{\delta}$, the random error goes up
from $2.6\,\rm{mas/yr}$ for quasars brighter than $u=18\,\rm{mag}$ to
$7.1\,\rm{mas/yr}$ for quasars fainter than $u=22\,\rm{mag}$. The increasing
of the random error may be caused by the large measuring errors of the fainter
sources on image. $\sigma_{\mu}=a+b\times\exp^{(-m/c)}$ is used to describe
the random error of proper motions as a function of magnitude, where
$\sigma_{\mu}$ is the random error of proper motions, $a$, $b$, and $c$ are
the unknown parameters to be fitted, and $m$ is the SCUSS $u$-band magnitude.
Table 2 lists the best fit parameters for proper motions in right ascension
and declination.

Figure 6: The scatter of proper motions $\mu_{\alpha}$ cos $\delta$ (upper)
and $\mu_{\delta}$ (lower) for SDSS spectroscopically confirmed quasars in
SCUSS catalog as a function of the SCUSS $u$-band magnitude from 16 to
$24\,\rm{mag}$. The systematic and random errors for quasar groups are showed
as red points and red error bars.

The scatters of $\mu_{\alpha}$cos$\delta$ (upper) and $\mu_{\delta}$ (lower)
of quasars as a function of SDSS $g-r$ color are shown in Figure 7. Quasars
with $g-r$ color of between $-0.25$ and 1.75 are divided into different color
groups with a group size of 0.3. There is no relation between error of proper
motions and color. The absolute values of systematic errors of
$\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$ are around $0.1\,\rm{mas/yr}$.
The random errors of $\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$ are both
about $4.0\,\rm{mas/yr}$. The CdE of SCUSS proper motion catalog is well
eliminated. The random error of SCUSS proper motion catalog does not
significantly change with color.

Figure 7: The scatter of proper motions $\mu_{\alpha}$ cos $\delta$ (upper)
and $\mu_{\delta}$ (lower) for SDSS spectroscopically confirmed quasars in
SCUSS catalog as a function of the SDSS $g-r$ color from $-0.25$ to 1.75. The
systematic and random errors for quasar groups are showed as red points and
red error bars.

##  5 COMPARISON OF PROPER MOTIONS BETWEEN SCUSS AND SDSS CATALOG

Comparing the SCUSS proper motion catalog with other catalogs can estimate the
external error of proper motions. The SDSS proper motion catalog (Munn et al.
2004, 2008; Ahn et al. 2012) covers most of the SCUSS fields. We compare the
proper motions between SCUSS and SDSS catalog. The SDSS proper motion catalog
is derived by combining SDSS and re-calibrated USNO-B astrometry. For SDSS
proper motion catalog, the random errors in right ascension and declination
are about $4.2\,\rm{mas/yr}$ and $3.7\,\rm{mas/yr}$, respectively (Munn et al.
2004). The median proper motion for all the quasar sample is about
$0.2\,\rm{mas/yr}$, but the systematic errors can be larger by a factor of 2 -
3 in small sky patches (Bond et al. 2010; Ahn et al. 2012).

The common stars between SCUSS catalog and SDSS catalog are eslected and
divided into different position ($\alpha$, $\delta$) and magnitude groups. The
means and the dispersions of differences of proper motions in different common
star groups are derived. The mean differences of proper motions as a function
of $\alpha$ are plotted in Figure 8. The upper panel is the mean differences
of $\mu_{\alpha}$ of stars and the lower panel is the mean differences of
$\mu_{\delta}$ of stars. Figure 8 shows that, for many regions, the mean
differences of proper motions between SDSS and SCUSS are small. From the upper
panel of Figure 8, we can see that, in the region
$60\textordmasculine<\alpha<70\textordmasculine$, the absolute values of mean
differences of $\mu_{\alpha}$ of common stars are about $1.5\,\rm{mas/yr}$.
The lower panel shows that, in the region
$0\textordmasculine<\alpha<10\textordmasculine$,
$60\textordmasculine<\alpha<70\textordmasculine$,
$310\textordmasculine<\alpha<320\textordmasculine$ and
$320\textordmasculine<\alpha<330\textordmasculine$, the absolute values of
mean differences of $\mu_{\delta}$ of common stars are about
$1.0\,\rm{mas/yr}$. However, in the region $\alpha=320\textordmasculine$, the
absolute value of mean difference of $\mu_{\delta}$ of stars reaches about
$3.5\,\rm{mas/yr}$. In most regions, the mean differences of the proper
motions are smaller than the random errors of SDSS and SCUSS proper motions.
The means and dispersions of differences of proper motions as a function of
$\delta$ are plotted in Figure 9. From Figure 9, we can see that the mean
differences of proper motions between SDSS and SCUSS are small. The absolute
values of mean differences are less than $0.5\,\rm{mas/yr}$.

The means and dispersions of differences of proper motions as a function of
SCUSS $u$-band magnitude are plotted in Figure 10. Figure 10 shows that, in
the magnitude range $u>17\,\rm{mag}$, the mean differences of proper motions
between SDSS and SCUSS are small and the dispersion of the differences of
proper motions increases with magnitude.

Figure 8:  The mean differences of proper motions $\mu_{\alpha}$ (upper) and
$\mu_{\delta}$ (lower) of common stars between SDSS and SCUSS as a function of
$\alpha$. The typical dispersion of differences of proper motions is about
$5\,\rm{mas/yr}$.

Figure 9:  The mean differences of proper motions $\mu_{\alpha}$ (upper) and
$\mu_{\delta}$ (lower) of common stars between SDSS and SCUSS as a function of
$\delta$. The typical dispersion of differences of proper motions is about
$5\,\rm{mas/yr}$.

Figure 10: The means and dispersions of differences of proper motions
$\mu_{\alpha}$ (upper) and $\mu_{\delta}$ (lower) of common stars between SDSS
and SCUSS as a function of the SCUSS $u$-band magnitude.

##  6 CONCLUSION

The absolute proper motions of SCUSS program are computed by combining the
SCUSS and GSC-II. The basic method of calculation of absolute proper motions
follows that of APOP. The measured coordinates ($x$, $y$) of objects instead
of equatorial coordinates are used to calculate the absolute proper motions.
The stellar sources with good imaging quality are used to eliminate the PdE
between the reference and program image. The galaxies with good imaging
quality are used to eliminate the MdE, CdE and the residual PdE.

The 22831 common quasars among SCUSS and SDSS are selected to test the
properties of SCUSS proper motion catalog. The systematic errors of the proper
motions are $-0.08$ and $-0.06\,\rm{mas/yr}$ in $\mu_{\alpha}$cos$\delta$ and
$\mu_{\delta}$. In addition, the random errors of the proper motions are 4.2
and $4.4\,\rm{mas/yr}$ in $\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$. We
made a detailed analysis of the obtained proper motions in order to determine
the position-, magnitude- and color- dependent errors of the proper motions.
Finally, we find that the PdE, MdE and CdE of SCUSS proper motion catalog are
small. The position- and color- dependent random errors remain about the same.
The random errors of $\mu_{\alpha}$cos$\delta$ and $\mu_{\delta}$ increase
with magnitude. Beyond that, no significant relations between errors
(systematic and random) and the position, magnitude and color are found.

The common stars between SCUSS and SDSS catalog are selected. The means and
dispersions of differences of proper motions between SCUSS and SDSS are
derived. The mean differences of proper motions between SDSS and SCUSS as a
function of $\alpha$ are small except for some regions. There is no relation
between the mean differences of proper motions and $\delta$. The mean
differences of proper motions are small in magnitude region $u>17\,\rm{mag}$.

The SCUSS proper motion catalog with small systematic error and low random
error can be used for many astronomical studies such as the Galaxy disk
kinematics, substructure of the Galaxy and Galaxy rotation curve.

The release of the SCUSS proper motion catalog is in preparation. The final
version of the SCUSS proper motion catalog will be available when the SCUSS
data becomes public. If you need a preliminary version of SCUSS proper motion
catalog, you can email Xiyan Peng (xypeng@bao.ac.cn).

## ACKNOWLEDGMENTS

We would like to thank the anonymous referee for providing rapid and
thoughtful report that helped improve the original manuscript greatly. This
work is supported by National Basic Research Program of China (973 Program),
No.2014CB845702, No. 2014CB845704, and No. 2013CB834902. This work is also
supported by the National Natural Science Foundation of China (NSFC,
Nos.11433005, 11373035, 11373033, 11373003, 11273003, 11203034, 11203031,
11303038, 11303043). The project is managed by the National Astronomical
Observatory of China and Shanghai Astronomical Observatory. This work is also
supported by the joint fund of Astronomy of the National Nature Science
Foundation of China and the Chinese Academy of Science, under Grants U1231113.
The SCUSS is funded by the Main Direction Program of Knowledge Innovation of
Chinese Academy of Sciences (No. KJCX2-EW-T06). It is also an international
cooperative project between National Astronomical Observatories, Chinese
Academy of Sciences and Steward Observatory, University of Arizona, USA.
Technical supports and observational assistances of the Bok telescope are
provided by Steward Observatory.

## References

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  * Zhou et al. (2014) Zhou, X., et al. 2014, RAA, in preparation

Table 1: Plate Information from GSC-II  Survey | Epoch | Band | Depth | plates used | sky coverage
---|---|---|---|---|---
Pal-QV | 1983-85 | $V_{12}$ | 19.5 | 103 | $\delta\geq 0^{\circ}$
SERC-EJ | 1979-88 | $B_{J}$ | 23.0 | 72 | $-15<\delta\leq 0$
POSS-$I$ E | 1950-1958 | $E$ | 20.0 | 144 | $\delta\geq-30$
POSS-$II$ O | 1950-1958 | $O$ | 21.0 | 143 | $\delta\geq-30$
POSS-$II$ J | 1987-00 | $B_{J}$ | 22.5 | 141 | $\delta\geq 0$
POSS-$II$ F | 1987-99 | $R_{F}$ | 20.8 | 141 | $\delta\geq 0$
POSS-$II$ N | 1989-02 | $I_{N}$ | 19.5 | 139 | $\delta\geq 0$
SERC ER | 1990-98 | $R_{F}$ | 22.0 | 93 | $-15<\delta\leq 0$
SERC I | 1990-02 | $I_{N}$ | 19.5 | 93 | $\delta\leq 0$
Table 2: Best-Fitting Parameters to Describe the Magnitude Dependence of the Random Errors of Proper Motions Type of $\sigma_{\mu}$ | a | b | c
---|---|---|---
$\mu_{\alpha}$ cos $\delta$ | $-4.78$ | 1.08 | $-9.47$
$\mu_{\delta}$ | $-4.77$ | 1.23 | $-10.07$


ESFRAD. FORTRAN code for calculation of QED corrections to polarized ep-
scattering
by the electron structure function method

A. Afanaseva), I. Akushevichb), A. Ilyichevc), N. Merenkovd)

(a)Jefferson Lab, Newport News, VA 23606, USA

(b)Duke University, Durham, NC 27708, USA

(c) National Center of Particle and High Energy Physics, 220040 Minsk, Belarus

(d) NSC ”Kharkov Institute of Physics and Technology”,

61108 Kharkov, Ukraine

###### Abstract

The main features of the electron structure function method for calculations
of the higher order QED radiative effects to polarized deep-inelastic
$ep$-scattering are presented. A new FORTRAN code ESFRAD based on this method
was developed. A detailed quantitative comparison between the results of
ESFRAD and other methods implemented in the codes POLRAD and RADGEN for
calculation of the higher order radiative corrections is performed.

##  1 Introduction

The observed cross section in the electron–proton deep inelastic scattering

$e^{-}(k_{1})+p(p_{1})\to e^{-}(k_{2})+X(p_{x})$ (1)

($k_{1}^{2}=k_{2}^{2}=m^{2}$) is affected by electromagnetic effects caused by
real and virtual photon emission and $e^{+}e^{-}$ – pair production, which
cannot be removed with any experimental cuts. Therefore contributions of these
effects called radiative corrections (RC), to different observables has to be
calculated theoretically and then subtracted from experimental data (see [1]
for details).

In the present report the model–independent RC are considered. Such RC include
contributions due to radiation of real but not observed photons and
electron–positron pairs as well as loop corrections in the lepton part of
interaction. The model–dependent RC arise from electromagnetic effects in the
hadron part of interaction. Such RC are described by the box–type and proton
vertex diagrams and real photon radiation by the proton. Their calculation
requires additional assumptions about the hadron structure and, therefore,
they have additional theoretical uncertainties. The model–dependent RC are the
subject of separate theoretical studies (see, e.g, Ref. [2, 3, 4, 5] for the
elastic $ep$-scattering case).

The structure of model–independent RC in ultrarelativistic approximation has a
simple form. The contribution of virtual and real particles to the observed
cross section in the $n$-th order of perturbation theory can be presented as
the following sum,

$\frac{d\sigma_{n}^{RC}}{dQ^{2}dy}=\left(\frac{\alpha}{\pi}\right)^{n}\sum_{i=0}^{n}C_{ni}(Q^{2},y)L^{i}+{\cal
O}(\frac{m^{2}}{Q^{2}}),\;$ (2)

where $Q^{2}=-(k_{1}-k_{2})^{2}$, $y=(k_{1}p_{1}-k_{2}p_{1})/k_{1}p_{1}$,
$L=\log Q^{2}/m^{2}$ and the coefficients $C_{ni}(Q^{2},y)$ are independent on
the electron mass. Each of the leading terms $C_{ni}(Q^{2},y)L^{i}$ describes
along with elastic (i.e., without no emitted photons) and inelastic events
with the number of hard collinear photons up to $n$. Since at modern
accelerator energies the quantity $L$ is about 10 - 20, the main contribution
to RC (2) gives so-called leading order (LO) terms which are proportional to
$(\alpha L/\pi)^{n}$ and all the real photons as well as $e^{+}e^{-}$-pairs in
this case are collinear of the parental electrons. Note that the contribution
$\sim(\alpha/\pi)^{n}L^{n-1}$ is usually called next-to-leading order (NLO).

At present only the lowest order radiated corrections ($n=1$) have been
calculated completely for inelastic [6] and elastic lepton-hadron scattering
[7] using the covariant approach developed in [8]. The formalism of [6] was
used later to develop a FORTRAN code POLRAD [9] and a Monte Carlo generator
RADGEN [10] for simulation of radiative events in deep–inelastic
$ep$–scattering.

Unfortunately, evaluation of the higher order effects with $n\geq 2$ were
estimated not so precise as lowest ones because of the extremely complicated
integration over the unobserved real photon phase space. Firstly, the soft
photon approximation was used by by D. Yennie, S. Frauchi and H. Suura [11] to
describe the multi–photon radiation. Then the calculation of RC within the
electron structure function (ESF) method was proposed by Fadin and Kuraev [12]
who applied the Drell–Yan representation for the electron–positron
annihilation cross section. According to this method, the radiatively
corrected observed cross section can be written as a convolution of two ESF
with its hard part. Within the leading accuracy the hard part coincides with
the Born cross section.

Further development of the ESF method was done in [13]. It consists in
modification of the hard part of the cross section taking into account the
possibility of one non–collinear hard photon radiation. Such modification
allows to go beyond the leading approximation and to compute partly NLO
contributions into RC of the order $(\alpha/\pi)^{n}L^{(n-1)}$

$\frac{d\sigma_{LL}^{RC}}{dQ^{2}dy}=\sum_{n=1}^{\infty}\left(\frac{\alpha}{\pi}L\right)^{n}C_{nn}(Q^{2},y)+\left(\frac{\alpha}{\pi}\right)^{n}L^{(n-1)}C_{nn-1}(Q^{2},y),$
(3)

while the NLO contribution of the order $\alpha/\pi$ is calculated exactly.
There exists another source of the NLO terms like the second one in the
right–hand side of Eq.(3), which could not be evaluated in [13]. Such terms
arise due to the improvement of ESFs by including their non-leading pieces.
The respective calculations are in progress now.

The second order correction ($\sim(\alpha L/\pi)^{2}$) for unpolarized and
polarized inelastic lepton-hadron scattering was evaluated numerically in [14]
and [15], respectively, just the ESF method. In this report we analyze the
results of the new FORTRAN code ESFRAD [16] based on approach developed in
[13] for both unpolarized and polarized deep-inelastic scattering and compare
it with the codes POLRAD and RADGEN that treat the higher–order QED effects
differently.

##  2 Master Formula

A master formula for the observed cross section within the ESF method is
derived according to Fig. 1 that should be considered as a diagram for the
cross section (not for the amplitude),

$\frac{d^{2}\sigma^{obs}(k_{1},k_{2})}{dQ^{2}dy}=\int\limits^{1}_{z_{1m}}dz_{1}\int\limits^{1}_{z_{2m}}dz_{2}D(z_{1},L)\frac{1}{z_{2}^{2}}D(z_{2},L)\frac{d\sigma^{hard}(\tilde{k}_{1},\tilde{k}_{2})}{d\tilde{Q}^{2}d\tilde{y}},\;$
(4)

where the physical meaning of integration variables $z_{1}$ and $z_{2}$ can be
understood in Fig.1. The hard cross section in the integrand depends on
so–called shifted variables which are defined as follows

$\tilde{k}_{1}=z_{1}k_{1},\;\tilde{k}_{2}=\frac{k_{2}}{z_{2}},\;\tilde{Q}^{2}=\frac{z_{1}}{z_{2}}Q^{2},\;\tilde{y}=1-\frac{1-y}{z_{1}z_{2}}.$
(5)

Figure 1: Diagram for the corrected cross section within the ESF method

The lowest limits of integration can be found as a solution of the standard
$\pi$-production threshold inequality for DIS process:
$(p_{1}+\tilde{k}_{1}-\tilde{k}_{2})^{2}>(M+m_{\pi})^{2}$ that leads to

$z_{2m}=\frac{1-y+xyz_{1}}{z_{1}-z_{th}},\;z_{1m}=\frac{1-y+z_{th}}{1-xy},\;z_{th}=\frac{m_{\pi}(m_{\pi}+2M)}{2p_{1}k_{1}}$
(6)

The electron structure function $D(z_{1},L)$ gives the energy fraction
$(z_{1})$ distribution of electrons with virtuality up to $Q^{2}$ which is
created by the initial electron, whereas $D(z_{2},L)$ is in fact the
fragmentation function of the heavy intermediate electron into real one with
the energy fraction $z_{2}.$ Function $D(z,L)$ absorbs all the leading
contributions connected with real and virtual photon emission and pair
productions

$D(z,L)=D^{\gamma}(z,L)+D^{e^{+}e^{-}}_{N}(z,L)+D^{e^{+}e^{-}}_{S}(z,L),\;\int\limits_{0}^{1}dz[D^{\gamma}(z,L)+D_{N}^{e^{+}e^{-}}(z,L)]=1.$
(7)

Roughly speaking, the contribution to the function $D^{\gamma}$ comes from all
the graphs with radiation of real and virtual photons, which is represented
here in symbolic form as follows

A nonsinglet part of $D^{e^{+}e^{-}}$ is described in the same graphs but with
attached $e^{+}e^{-}$-pairs to the photon lines.

The graphs that have at least one intermediate photon which propagates along
the parental electron line contribute to the singlet part of $D^{e^{+}e^{-}}$.

For details about the $D$-function see, for example, [17].

The hard cross section in the master formula consists of the sum of the Born
cross section and NLO part of lowest order RC. It can be written as

$\displaystyle\frac{d\sigma^{hard}(k_{1},k_{2})}{dQ^{2}dy}$ $\displaystyle=$
$\displaystyle\frac{1}{[1-\alpha/\pi\Pi(Q^{2})]^{2}}\left[\frac{d\sigma_{0}(k_{1},k_{2})}{dQ^{2}dy}+\left(\frac{\alpha}{\pi}\right)C_{10}(Q^{2},y)\right],$
(8)

where $\Pi(Q^{2})$ is an intermediate photon self energy and the coefficient
$C_{10}(Q^{2},y)$ enters Eq.(2).

The explicit expressions for the electron structure functions, used in ESFRAD
as well as the more detailed description of the hard cross section can be
found in [13, 16].

##  3 Numerical Results

For the numerical analysis performed within experiments on fixed proton
targets at JLab ($E_{beam}=4$ GeV) and HERMES ($E_{beam}=27.5$ GeV), NMC
parametrization for the unpolarized hadron structure function $F_{2}$ is used.
The polarized hadron structure functions $g_{1}$ and $g_{2}$ are expressed by
the asymmetries

$A_{1}=a+x^{b}(1-\exp(cx)),\;$ (9)

with $a=1.90202\times 10^{-2}$, $b=-1.16312\times 10^{-3}$, $c=-1.8451$ and

$A_{2}=\frac{0.53Mx}{\sqrt{Q^{2}}}$ (10)

in a standard way.

The $y$-dependences of the relative RC to unpolarized part of the cross
section

$\delta_{u}=\frac{d\sigma^{obs}/dQ^{2}dy-d\sigma^{0}/dQ^{2}dy}{d\sigma^{0}/dQ^{2}dy},$
(11)

calculated in different FORTRAN codes and methods are presented on Fig. 2. It
should be noted that the lowest order RC are the same for POLRAD, RADGEN and
ESFRAD. But the higher order corrections in these codes are estimated in
different way:

  * •

In RADGEN the soft photon approximation is used to account multi–photon
events.

  * •

In POLRAD only the second order leading correction is included.

  * •

In ESFRAD together with the leading contribution the next-to-leading one is
partly incorporated in accordance with Eq.(3).

It can be seen from Fig. 2(a) that at HERMES kinematics POLRAD and ESFRAD give
very similar results for unpolarized part of the cross section while the
behavior of the higher order correction generated by RADGEN coincides with
POLRAD and ESFRAD only at large values of the variable $y.$ At the same time
at JLab kinematics (see Fig. 2(b)) the higher order effects in the considered
different approaches show more distinctions. In this case the results of
POLRAD and ESFRAD agree only at low $y$ and there is noticeable disagreement
between them when $y$ increases and approaches its maximum value. On the other
hand, RADGEN disagrees with POLRAD and ESFRAD at low $y$ and coincides with
POLRAD at large $y.$ Note that for any given kinematical point $(x,y)$ the
value of RC for JLab is shifted toward positive values compared to the
corresponding RC for HERMES.

Radiative corrections to the polarized asymmetries (9), (10) are presented in
Fig. 3 and Fig. 4, respectively. One can see that the higher order effects
give a very small contribution to the total correction in the entire kinematic
region. It means that the effective higher order correction has a rather
factorized form and disappears in the ratio of the polarization–dependent and
unpolarized parts of the cross section. It should be noted that the main
difference in the behavior of RC to $A_{1}$ and $A_{2}$: when $x$ is growing
RC to $A_{1}$ increases while RC to $A_{2}$ decreases. As well as for
unpolarized case the value of RC to asymmetries for JLab is larger than the
corresponding RC for HERMES.

$a)$ | $b)$
---|---
Figure 2: Relative radiative corrections to the unpolarized cross section
defined by formulae (11) for a) HERMES ($E_{beam}=27.57$ GeV) and b) JLab
($E_{beam}=4$ GeV) kinematic conditions

##  4 Conclusion

In this report the electron structure function method for calculation of the
higher order radiative corrections is overviewed briefly and a comparative
analysis of the FORTRAN code ESFRAD developed on the basis of ESF approach,
with POLRAD and RADGEN is performed. All these codes account exactly for the
first–order correction but the higher order effects are incorporated into them
in different ways.

Our numerical analysis of the higher order RC in deep-inelastic events shown
that

  * •

RC to the unpolarized cross section and the spin asymmetries at JLab
kinematics conditions ($E_{beam}=4$ GeV) are larger than at HERMES ones
($E_{beam}=27.5$ GeV).

  * •

The higher order corrections generated by ESFRAD and POLRAD have the similar
behavior at HERMES but differ noticeably at JLab conditions.

  * •

At large values of $y$ where RC is very important all three codes give the
same result at HERMES kinematics but at JLab the higher order contribution of
ESFRAD decreases considerably the total RC as compared with POLRAD and RADGEN.

  * •

The higher order corrections contribute very little to polarization
asymmetries that means that effectively they have rather factorized form and
practically are canceled in the ratio of unpolarized and polarized parts of
the cross section in the entire kinematic region allowed by HERMES and JLab
conditions.

  * •

With the increasing $x$ the total RC to asymmetry $A_{1}$ increases, while RC
to asymmetry $A_{2}$ decreases at both HERMES and JLab experimental
conditions.

$a)$ | $b)$
---|---
Figure 3: Radiative corrections to $A_{1}$ defined by expression (9) for a) HERMES ($E_{beam}=27.57$ GeV) and b) JLab ($E_{beam}=4$ GeV) kinematic conditions $a)$ | $b)$
---|---
Figure 4: Radiative corrections to $A_{2}$ defined by expression (10) for a)
HERMES ($E_{beam}=27.57$ GeV) and b) JLab ($E_{beam}=4$ GeV) kinematic
conditions

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