The Title of the Paper

I. Author1 example@list.ru Institute/Organization/Place of Job, City/Town, ZIP-code Country I. Author2 Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnii Arkhyz, 369167 Russia

Abstract

We study the nearest outskirts ( $R<3R_{200c}$ ) of 40 groups and clusters of galaxies of the local Universe ( $0.02<z<0.045$ and 300 km s ${}^{-1}<\sigma<950$ km s^-1). Using the SDSS DR10 catalog data, we determined the stellar mass of galaxy clusters corresponding to $K_{s}$ -luminosity (which we determined earlier based on the 2MASX catalog data) $(M_{*}/M_{\odot})\propto(L_{K}/L_{\odot})^{1.010\pm 0.004}$ ( $M_{K}<-21\hbox{$\,.\!\!^{\rm m}$}5$ , $R<R_{200c}$ ). We also found the dependence of the galaxy cluster stellar mass on halo mass: $(M_{*}/M_{\odot})\propto(M_{200c}/M_{\odot})^{0.77\pm 0.01}$ …

stars: double or binary—stars: individual: ADS 48

1 INTRODUCTION

The multiple star system ADS 48, discovered by Otto Struve in 1876, has been repeatedly investigated by various authors (see, for example, G-L, Hopman), but their attention was mainly attracted by the inner pair AB. According to the identification in the WDS catalog, the three stars—A, B and F—are physically connected (by common parallax and proper motions). For the inner AB pair, there has been a long series of observations since its discovery, and the F component has not been observed for almost a century.

2 FIRST SECTION

Here comes some math:

	$\displaystyle\rho$	$\displaystyle=$	$\displaystyle\sqrt{x^{2}+y^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \,\qquad\theta\;=\;\arctan\dfrac{x}{y}$
	$\displaystyle\mu$	$\displaystyle=$	$\displaystyle\sqrt{\mu_{x}^{\prime 2}+\mu_{y}^{\prime 2}}\leavevmode\nobreak\ ,\qquad\psi\;=\;\arctan\dfrac{\mu_{x}^{\prime}}{\mu_{y}^{\prime}}.$

Here

$\displaystyle x$	$\displaystyle=$	$\displaystyle(\alpha_{\rm B}-\alpha_{\rm A})\cos{\delta}\times 3600,$
$\displaystyle y$	$\displaystyle=$	$\displaystyle(\delta_{\rm B}-\delta_{\rm A})\times 3600,$
$\displaystyle\delta$	$\displaystyle=$	$\displaystyle(\delta_{\rm A}+\delta_{\rm B})/2,$
$\displaystyle\mu_{x}^{\prime}$	$\displaystyle=$	$\displaystyle\mu_{x{\rm B}}-\mu_{x{\rm A}},\qquad\mu_{y}^{\prime}=\mu_{y{\rm B}}-\mu_{y{\rm A}}.$

Table 1: Two column table

pair	AB	AB	AB–F	AB–F	AB–F
Instrument	$26^{\prime\prime}$ , CCD	GAIA	$26\hbox{${}^{\prime\prime}$}$ , photo	$26\hbox{${}^{\prime\prime}$}$ , photo	GAIA
Instrument	individual	GAIA	individual	smoothed	GAIA
Parameters
$T_{1}-T_{2}$	$2003$ – $2012$	–	$1968$ – $1995$	1971–1992	–
$T_{0}$	$2008.6$	$2015.5$	$1981.5$	1981.5	$2015.5$
n	$48$	–	$115$	30	–
$\rho$ , arcsec.	$6.0534$	$6.00768^{*}$	$327.3322$	$327.3339$	$327.1754$
	$\pm 0.0012$	$\pm 0.00008$	$\pm 0.0023$	$\pm 0.0010$	$\pm 0.0002$
$\theta_{2000}$ , degr.	$185.3604$	$188.2084$	$254.2942$	$254.2943$	$254.25739$
	$\pm 0.0059$	$\pm 0.0010$	$\pm 0.0017$	$\pm 0.0005$	$\pm 0.00001$
$\mu$ , mas/year	$43.1$	$44.94$	$4.3$	$3.9$	$5.4$
	$\pm 0.3$	$\pm 0.18$	$\pm 0.5$	$\pm 0.2$	$\pm 0.1$
$\psi_{2000}$ , degr.	$283.09$	$288.06$	$86.4$	$73.2$	$37.8$
	$\pm 0.7$	$\pm 0.16$	$\pm 19.6$	$\pm 7.4$	$\pm 1.1$
$\dot{\rho}$ , mas/year	$-6.4$	$-7.7$	$-4.2$	$-3.9$	$-4.4$
	$\pm 0.5$	$\pm 0.1$	$\pm 0.4$	$\pm 0.2$	$\pm 0.1$
$\dot{\theta}$ , degr./year	$0.4034$	$0.4202$	$-0.0002$	$-0.0000$	$-0.00056$
	$\pm 0.0024$	$\pm 0.0017$	$\pm 0.0003$	$\pm 0.0001$	$\pm 0.00001$
Here $n$ is the number of individual or smoothed observations,
^∗—the value $\rho$ is given adjusted for Gaia–CCD $=+0\hbox{$\,.\!\!^{\prime\prime}$}03$ .

Refer to caption — Figure 1: Two column figure.

Table 1 shows AMPs calculated from observations of Gaia DR2, and long-term series of observations of the Pulkovo 26-inch refractor. For the AB pair, we compare only with the CCD observations of 2003–2012. For this pair, we found a systematic discrepancy in $\rho$ , which is clearly visible in Fig. 1.

3 ANOTHER SECTION

The text of the Section.

4 THIRD SECTION

The motion of the outer pair is in the direction $\rho$ , and we can definitely state that for all orbits of the family, the inclination of the orbit $i\approx 90\hbox{${}^{\circ}$}$ , and the longitude of the ascending node $\Omega\approx\theta-180\hbox{${}^{\circ}$}$ . Therefore, you can calculate the angle between the planes of the outer and inner orbits. As a result, we get that the planes of the orbits are non-planar.

Equations:

v_{1}=\sqrt{\dfrac{4\pi^{2}{m_{2}}^{2}}{r(m_{1}+m_{2})}}.

(1)

Assuming $m_{2}<<m_{1}$ , we get

m_{2}=v_{1}\times\sqrt{\dfrac{m_{1}}{4\pi^{2}}r},

(2)

\overrightarrow{v_{1}}=f\times(\overrightarrow{\mu_{\rm G}}-\overrightarrow{\mu_{\rm ph}})/p_{t},

(3)

where $\overrightarrow{\mu_{\rm ph}}=(\mu_{\rm ph}\sin\,\psi_{\rm ph},\mu_{\rm ph}\cos\,\psi_{\rm ph})$ is the average orbital motion obtained from a long series of photographic observations; $\overrightarrow{\mu_{\rm G}}$ is the instantaneous orbital motion determined by Gaia observation; $p_{t}=87$ mas is the parallax; $f$ is the coefficient of transition from the relative velocity of the orbital motion to the velocity relative to the center of mass of the hierarchical triple system, which is motionless. If component F has a satellite, then

f_{\rm F}=M_{\rm A+B}/M_{\rm A+B+F}.

If the center of mass of the AB system oscillates, then

$\displaystyle f_{\rm C}$	$\displaystyle=$	$\displaystyle M_{\rm F}/M_{\rm A+B+F},$
$\displaystyle f_{\rm A}$	$\displaystyle=$	$\displaystyle f_{\rm C}\,(M_{\rm A+B}/M_{\rm A}),$
$\displaystyle f_{\rm B}$	$\displaystyle=$	$\displaystyle f_{\rm C}\,(M_{\rm A+B}/M_{\rm B}).$

If we use the values of $\mu_{ph}$ according to the smoothed series, then

	$\displaystyle m_{2,{\rm F}}/\sqrt{r}$	$\displaystyle=$	$\displaystyle 0.0030\pm 0.0006\leavevmode\nobreak\ M_{\odot},$
	$\displaystyle m_{2,{\rm A}}/\sqrt{r}$	$\displaystyle=$	$\displaystyle m_{2,{\rm B}}/\sqrt{r}=0.0027\pm 0.0006\leavevmode\nobreak\ M_{\odot};$

if we use the values of $\mu_{ph}$ according to individual observations, then

	$\displaystyle m_{2,{\rm F}}/\sqrt{r}$	$\displaystyle=$	$\displaystyle 0.0039\pm 0.0017\leavevmode\nobreak\ M_{\odot},$
	$\displaystyle m_{2,{\rm A}}/\sqrt{r}$	$\displaystyle=$	$\displaystyle m_{2,{\rm B}}/\sqrt{r}=0.0035\pm 0.0015\leavevmode\nobreak\ M_{\odot}.$

Table 2: One column table.

Parameters	Orbits
Parameters	1	2	3	4
$a_{\rm ph}$ , mas	15.0	14.3	8.2	4.0
$P$ , year	11.0	11.04	9.52	10.97
$e$	0.2	0.24	0.53	0.3
$i$ , degr.	97.0	96.3	179.98	44
$\omega$ , degr.	235.0	258.6	79.8	56.2
$\Omega$ , degr.	147.2	143.2	12.0	217.1
$T$ , year	1980.0	1980.56	1988.15	1982.8
$V_{r\gamma}$ , m c^-1	–	–	$-0.7$	–
$p_{t}$ , mas	87.0	87.0	87.0	86.9
$M_{1},\leavevmode\nobreak\ M_{\odot}$	0.5	0.5	0.5	0.65
$a_{1}$ , a.u.	0.17	0.16	0.094	0.046
$M_{2},\leavevmode\nobreak\ M_{\odot}$	0.023	0.022	0.013	0.007
$a_{2}$ , a.u.	3.82	3.82	3.50	4.28
$\sigma_{x}$ , mas	2.2	2.0	3.6	5.1
$\sigma_{y}$ , mas	12.3	12.0	13.5	5.4
$\sigma_{V_{r}}$ , mas/year	–	–	0.078	–

The system of equations is solved:

x(t)=x_{0}+\dot{x}(t-t_{0})+BX_{\varphi}+GY_{\varphi},

(4)

y(t)=y_{0}+\dot{y}(t-t_{0})+AX_{\varphi}+FY_{\varphi},

(5)

where $x=\rho\sin\theta$ , $y=\rho\cos\theta$ ; phase $\varphi=(t-t_{0})/P$ ; $X_{\varphi}=\cos\,(E_{\varphi})-e$ , $Y_{\varphi}=\sqrt{1-e^{2}}\sin\,(E_{\varphi})$ are orbital coordinates corresponding to dynamic orbital elements $P$ , $T$ and $e$ ; $x_{0}$ and $y_{0}$ are coordinates of the center of mass at the moment $t_{0}$ ; $A$ , $B$ , $F$ and $G$ are the Thiele-Innes elements, from which we obtain the geometric elements of the orbit ( $a$ , $i$ , $\omega$ , $\Omega$ ). In Table 2, this orbit is represented by number 2.

5 CONCLUSION

This paper demonstrates the possibility…

Acknowledgements.

The authors are grateful…

FUNDING

This work was supported by the…

CONFLICT OF INTEREST

The authors declare no conflicts of interest.