\section{Introduction}
The Bekenstein-Hawking entropies of the BPS saturated
black holes have been studied in four-, five-
and ten-dimensions from the nonperturbative aspects of string theory
\cite{CY,CT,SV,HS,MS,Tsey,GKT,BB}.
The black holes in ten-dimensions are constructed
by the intersecting D-branes.
The black holes are dual to each other
under the T-duality transformation.
The four- and five-dimensional black holes
are obtained by dimensional reduction
from the ten-dimensional black holes.
The method to obtain the non-extremal black holes
from the extremal black holes are studied.
Their entropies are interpreted in terms
of the microscopic states.
The effect of the graybody factor to the Hawking-radiate
blackbody radiation is also derived from the D-branes.

On the other hand, the Euler numbers of the black holes are
investigated in four-dimensions~\cite{HHR,GK,LP}.
The Euler number $\chi$ of an $n$-dimensional manifold
is the sum of the Betti numbers $B_p$:
\begin{eqnarray*}
 \chi = \sum_{p=0}^{n}(-1)^p B_p.
\end{eqnarray*}
The Betti numbers are integers.
Therefore the Euler numbers are also integers.
We can also calculate the Euler numbers by
using the $n$-dimensional Gauss-Bonnet action,
which integrates the curvatures $R_{ij}$
of the black holes.
The $n$-dimensional Gauss Bonnet action is
\begin{eqnarray*}
 \chi &=& \frac{1}{(4\pi)^{n} n!} \int dx^n
 \epsilon^{i_1i_2 \cdots i_n}
  R_{i_1i_2}R_{i_3i_4} \cdots R_{i_{n-1}i_n} .
\end{eqnarray*}
The black holes in four-dimensions have the period
in Euclidean time coordinate.
The period is defined by the inverse of the surface gravity.
By using this relation, we can obtain the integer valued Euler numbers.

In the same way as in four-dimensions,
we can calculate the Euler numbers of the black holes
in ten-dimensions using the ten-dimensional Gauss-Bonnet action.
The Euler numbers of the black holes in ten-dimensions
have not been studied so far.
We consider the Euler numbers
of the black holes which are realized by the intersecting D-branes
in ten-dimensions.
The black holes of the intersecting D-branes in ten-dimensions
also have the periods with respect to the compactification radii.
The radii are usually taken to be arbitrary.
For the generic radii,
the Euler numbers calculated
by the ten-dimensional Gauss-Bonnet actions are non-integers.
However the Euler numbers must be integers because of
their definition.
To avoid these difficulties, we propose a way to constrain
the compactification radii.

We reflect on the necessity
to fix the values of the compactfication radii
in the physical reasoning.
We treat the black holes in ten-dimensions.
These have six compactified directions.
The entropies of the black holes in ten-dimensions
are expressed by the compactification radii
and the charges.
We rewrite the entropies
using the quantized charges
and the Newton's constant,
then the entropies are irrespective
of the compactification radii.
Therefore it seems not to be necessary to fix
the value of the compactification radii.
However, the four-dimentional Newton's constant $G_{(4)}$
depends on the value of the compactified radii,
\begin{eqnarray*}
 G_{(4)} &=& G_{(10)}/L_1L_2L_3L_4L_5L_6 ,
\end{eqnarray*}
where $L_i$ are the compactification radii.
$G_{(10)}$ is the ten-dimensional Newton's constant,
\begin{eqnarray*}
 G_{(10)} = 8\pi^6 g^2
\end{eqnarray*}
with $\alpha' = 1$.
We find that
the compactified radii depend on
the radius of the horizon ($\mu$) and the charges,
using the method of the surface gravities
of the compactified directions,
discussed in the section 3.
Then the four-dimensional Newton's constant
is also rewritten by $\mu$ and the charges.
We are interested in the behavior of
the four-dimensional Newton's constant
in the BPS limit, $\mu \to 0$.
As a result, we obtain that
the Newton's constant vanishes in the BPS limit.

The another physical reasoning that we need to fix
the compactification radii
is to avoid the singular effects
of the horizon in the compactified directions.
We recall the way
to define the period in the time direction.
If no singular effects of the horizon exist in $t-r$ directions,
then the topology of these directions are ${\bf R^2}$,
and the Euler number of these directions is 1.
Using the Gauss-Bonnet theorem,
we find that
the Euclidean time coordinate
have the period which is
the inverse of the surface gravity.
Therefore we need to take this period
in the time direction
to avoid the singular effects of the horizon.
Similarly, we need to fix
the compactification radii to avoid the singular effects
of the horizon in the compactified directions.
We find that
the compactification radii are the inverses of
the surface gravity in the compactified directions
using the Gauss-Bonnet theorem.
We discuss this point in the section 3 .

The purpose of this paper is to constrain
the compactification radii
in order to obtain the integer valued Euler numbers
for the non-extremal black holes of the intersecting D-branes
in ten-dimensions.
We further study the microscopic interpretations of their
entropies.

We consider the black holes of the intersecting D-branes
which have some compactification radii
in ten-dimensions~\cite{CT,Tsey,GKT}.
We assume that the metrics only depend on
$r = \sqrt{x^2+y^2+z^2}$.
In the four-dimensions, the black holes have the period
in Euclidean time coordinate.
The period is defined by the inverse of the surface gravity.
We extend this construction to constrain the compactified radii
in ten-dimensions.
We use the inverse of the surface gravities
in the compactified
directions to constrain these radii.
As a result, we are able to obtain the integer Euler numbers.
We recall that the temperatures are defined by the surface gravities
in the time coordinates.
The relation between the radii and the surface gravities
generalizes the temperatures of the black holes.

Using these results,
we further obtain the entropies of the non-extremal
intersecting D-branes.
These entropies are invariant under T-duality transformation.
In the BPS limit,
we obtain the finite and non-vanishing entropies
only for two intersecting D-branes.
We observe that they can be interpreted as the product
of charges of each D-brane.
We then study the entropies of the boosted metrics.
We obtain the famous relation
between the entropies and the quantized D-brane charges
and the internal momenta of the intersecting D-branes.
These relations are the same as
that of the microscopic D-brane picture~\cite{SV,HS,MS}.

The organization of this paper is as follows.
In section 2, we review the way to construct the intersecting
D-branes of the type IIA and IIB superstrings.
We explain how to obtain the non-extremal black holes
from the extremal black holes.
In addition we define the boost transformation of the metrics.
In section 3, we consider the surface gravities
in the compactified directions
to constrain the compactification radii.
As a result, we obtain the Euler numbers which are integers.
In section 4, we calculate the entropies of the non-extremal
intersecting D-branes and study them in the BPS limit.
We further discuss the interpretation of these entropies
in terms of the microscopic states.

\section{Non-extremal Intersecting D-branes in ten-dimensions}
In this section we review the way
to construct the intersecting D-branes~\cite{Tsey}.
We then explain how to obtain non-extremal black holes
from the extremal black holes and to obtain the boosted metrics.
We treat the intersecting D-branes in ten-dimensions with the
three-dimensional transverse directions with respect to the D-branes.
This means that
the harmonic functions of these metrics only depend on
$r = \sqrt{x^2+y^2+z^2}$.

We introduce the metrics and the field strengths in association with
the D-branes.
The metrics of the D$n$-branes wrapped on the
$i_1, i_2, \cdots i_n$- directions are
\begin{eqnarray}
   ds^2_{10}  = H^{1/2}
    \big[ H^{-1}(-dt^2 + dx_{i_1}^2 + \cdots + dx_{i_n}^2 )
          + dr^2 + r^2 d\Omega^2_{10-n-2} \big] .
\end{eqnarray}
$H$ is the harmonic function.
The field strengths are
\begin{eqnarray*}
  {\cal F} = \left \{
\begin{array}{@{\,}ll}
 dt \wedge dH^{-1} \wedge dx_{i_1} \wedge
 \cdots \wedge dx_{i_n}
  & (n\le 3; \ electric) \\
 \ast(dt \wedge dH \wedge dx_{i_1} \wedge
 \cdots \wedge dx_{i_n})
  & (n\ge 3; \ magnetic) ,
\end{array}
\right.
\end{eqnarray*}
where $\ast$ is the Hodge dual in ten-dimensions.
In $n=3$, the field strength is self-dual.
Then they have the field strengths
both of electric and magnetic nature.
For the intersecting D-branes, ${\cal F}$ are
the sum of the field strengths of each D-brane.
We obtain the metrics of the type IIA string theory
from the intersecting M-branes by dimensional reductions.
The intersecting M-branes with the three-dimensional
transverse directions with respect to the M-branes
are $(2,2,5,5)_M$, $(5,5,5)_M$, $(2,5,5)_M$,
$(2,2,5)_M$, and $(5,5)_M$, where $(5,5)_M$ denote two
intersecting M5-branes, and so on~\cite{CT,Tsey,GKT}.

We subsequently review the metrics of three intersecting D-branes.
In ten-dimensions, we can obtain the (4,4,4) metric
from the intersecting M-branes $(5,5,5)_M$
upon dimensional reduction along a common direction
of three intersecting M5-branes :
\begin{eqnarray}
   ds^2_{10}  &=& (H_1 H_2 H_3)^{1/2}
    \big[ - H_1^{-1} H_2^{-1} H_3^{-1}dt^2 \nonumber \\
 &&   + H_1^{-1} H_2^{-1} (dx_1^2 + dx_2^2)
      + H_1^{-1} H_3^{-1} (dx_3^2 + dx_4^2) \nonumber \\
 &&   + H_2^{-1} H_3^{-1} (dx_5^2 + dx_6^2)
      + dr^2 + r^2 d\Omega^2_2 \big] .
\end{eqnarray}
The field strength ${{\cal F}_4}_{(4,4,4)}$
and the dilaton $\phi$ are
\begin{eqnarray}
 {{\cal F}_4}_{(4,4,4)}
 &=& (\partial_r H_1 \ d\theta \wedge d\phi \wedge dx_6 \wedge dx_7
  \nonumber \\
 && + \partial_r H_2 \ d\theta \wedge d\phi \wedge dx_4 \wedge dx_5
    + \partial_r H_3 \ d\theta \wedge d\phi \wedge dx_1 \wedge dx_2), \\
 e^{-2\phi} &=& (H_1H_2H_3)^{1/2},
\end{eqnarray}
where $H_i = 1+ \frac{Q_i}{r}$. We denote this metric
as the following~\cite{BB} :
\begin{eqnarray}
 (4,4,4) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
     \times & \times & \times & \times & \times & -  & -  & -  & -  & - & \\
     \times & \times & \times & -  & -  & \times & \times & -  & -  & - & \\
     \times & -  & -  & \times & \times & \times & \times & -  & -  & - & .
               \end{array} \right. $} } \label{444}
\end{eqnarray}

Similarly we can obtain the $(2,4,4)$ metric,
from $(2,5,5)_M$ upon dimensional reduction
along a direction of two intersecting M5-branes,
\begin{eqnarray}
 (2,4,4) &=& \mbox{ {\scriptsize
         $\left\{ \begin{array}{c|cccccccccc}
       \times & \times & \times & -  & -  & -  & -  & -  & -  & - & \\
       \times & \times & -  & \times & \times & \times & -  & -  & -  & - & \\
       \times & -  & \times & \times & \times & -  & \times & -  & -  & - & .
               \end{array} \right. $} }
\end{eqnarray}
The field strength ${{\cal F}_4}_{(2,4,4)}$
and the dilaton $\phi$ are
\begin{eqnarray}
 {{\cal F}_4}_{(2,4,4)}
 &=& ( dt \wedge \partial_r H_1^{-1} \ dr \wedge dx_1 \wedge dx_2
  \nonumber \\
 && + \partial_r H_2 \ d\theta \wedge d\phi \wedge dx_2 \wedge dx_6
    + \partial_r H_3 \ d\theta \wedge d\phi \wedge dx_1 \wedge dx_5), \\
 e^{-2\phi} &=& (H_1^{-1}H_2H_3)^{1/2}.
\end{eqnarray}

Along a direction of the M5-brane, we can obtain the $(2,2,4)$ metric,
\begin{eqnarray}
 (2,2,4) &=& \mbox{ {\scriptsize
         $\left\{ \begin{array}{c|cccccccccc}
      \times & \times & \times & -  & -  & -  & -  & -  & -  & - & \\
      \times & -  & -  & \times & \times & -  & -  & -  & -  & - & \\
      \times & \times & -  & \times & -  & \times & \times & -  & -  & - & .
               \end{array} \right. $} }
\end{eqnarray}
The field strength ${{\cal F}_4}_{(2,2,4)}$
and the dilaton $\phi$ are
\begin{eqnarray}
 {{\cal F}_4}_{(2,2,4)}
 &=& (dt \wedge \partial_r H_1^{-1} \ dr \wedge dx_1 \wedge dx_2
    \nonumber \\
 && +  dt \wedge \partial_r H_2^{-1} \ dr \wedge dx_3 \wedge dx_4
    + \partial_r H_3 \ d\theta \wedge d\phi \wedge dx_2 \wedge dx_4), \\
 e^{-2\phi} &=& (H_1H_2H_3^{-1})^{-1/2}.
\end{eqnarray}
In these constructions, we have regarded the D2-branes as electric
and the D4-branes as magnetic objects.

Moreover, we can obtain the remaining metrics
by T-duality transformation from these metrics
in the $i$-th direction~\cite{GKT} :
\begin{eqnarray*}
 g'_{ii} = 1/g_{ii}, \quad
 \exp(-2\phi') = g_{ii} \exp(-2\phi) .
\end{eqnarray*}
In our notation, the T-duality transformation
is realized by interchanging
$\times$ and $-$. For example, in the three intersecting D-branes case,
\begin{eqnarray}
 \mbox{ {\scriptsize
        $\left\{ \begin{array}{cc}
         \times & \\
         \times & \\
         -  &
 \end{array} \right. $} } \leftrightarrow
 \mbox{ {\scriptsize
        $\left\{ \begin{array}{cc}
         -  &  \\
         -  &  \\
         \times & .
                 \end{array} \right. $} }
\end{eqnarray}
If we apply the T-duality transformation
on the metric $(4,4,4)$ (in (\ref{444}))
in the first direction, this metric is changed to $(3,3,5)$.
Similarly, we can obtain the following metrics :
\begin{eqnarray*}
 && (4,4,4) \leftrightarrow
    (3,3,5) \leftrightarrow
    (2,4,4) \ or \ (2,2,6) \\
 &&  \leftrightarrow
    (3,3,3) \ or \ (1,3,5) \leftrightarrow
    (2,2,4) \ or \ (0,4,4) \\
 && \leftrightarrow
    (1,3,3) \leftrightarrow
    (2,2,2) .
\end{eqnarray*}
The metrics of the intersecting even-branes
are the metrics of type IIA.
The metrics of the intersecting odd-branes
are the metrics of type IIB.

For four intersecting D-branes, we obtain the following metric:
\begin{eqnarray}
 (2,2,4,4) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
     \times & \times & \times & -  & -  & -  & -  & -  & -  & - & \\
     \times & -  & -  & \times & \times & -  & -  & -  & -  & - & \\
     \times & \times & -  & \times & -  & \times & \times & -  & -  & - & \\
     \times & -  & \times & -  & \times & \times & \times & -  & -  & - & .
               \end{array} \right. $} }
\end{eqnarray}
from $(2,2,5,5)_M$.
Applying the T-duality transformation, we can obtain the following metrics :
\begin{eqnarray*}
 (3,3,3,3) \leftrightarrow (2,2,4,4) \leftrightarrow (1,3,3,5)
 \leftrightarrow (0,4,4,4) .
\end{eqnarray*}

For two intersecting D-branes, we obtain the $(4,4)$ metrics:
\begin{eqnarray*}
 (4,4) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
      \times & \times & \times & \times & \times &  - &  - & -  & -  & - & \\
      \times & \times & \times &  - &  - & \times & \times & -  & -  & - & ,
        \end{array} \right. $} }
\end{eqnarray*}
from $(5,5)_M$. Applying the T-duality transformation,
we can obtain following metrics :
\begin{eqnarray*}
 (4,4) \leftrightarrow (3,5) \leftrightarrow (2,6) , \\
 (3,3) \leftrightarrow (2,4) \leftrightarrow (1,5) , \\
 (2,2) \leftrightarrow (1,3) \leftrightarrow (0,4) ,
\end{eqnarray*}
in a non-intersecting direction, and
\begin{eqnarray*}
 (4,4) \leftrightarrow (3,3) \leftrightarrow (2,2) ,
\end{eqnarray*}
in an intersecting direction.
We have assumed that these metrics have the three-dimensional
transverse directions. Therefore all harmonic functions
in these metrics only depend on $r = \sqrt{x^2+y^2+z^2}$.
Note that our notation is different from~\cite{BB}.

These metrics can also be obtained from the metrics
of three intersecting D-branes. For example, we find the following
relation:
\begin{eqnarray*}
 (3,3,5) \to (3,3) ,
\end{eqnarray*}
by assuming the third D5-brane's charge to vanish.
For single D-brane, the T-duality transformation implies
the following relation
\begin{eqnarray*}
 (6) \leftrightarrow (5) \leftrightarrow
\cdots \leftrightarrow (0) .
\end{eqnarray*}

In order to obtain non-extremal metrics, we modify the above metrics as
follows~\cite{CT} :

(1) We make the following replacements in the transverse
space-time  part of the metric:
\begin{equation}
  dt^2  \to   f(r) dt^2 , \quad dr^2 \to f^{-1} (r) dr^2 ,
  \quad  f(r) = 1 - \frac{\mu}{r} .
\end{equation}
We also use the harmonic functions,
\begin{equation}
 H_i = 1 + \frac{{{\cal Q}}_i}{r} , \quad
{{\cal Q}}_i= \mu\sinh^2 \delta_i ,
\end{equation}
for the constituent D2-branes, and
\begin{equation}
 H_i = 1 + \frac{{{\cal P}}_i}{r} , \quad
{{\cal P}}_i= \mu\sinh^2 \gamma_i ,
\end{equation}
for the constituent D4-branes.

(2) In the expression for the field strength ${\cal F}_4$
of the three-form field, we make the following  replacements:
\begin{equation}
 H'_i \to H'_i = 1 + \frac{Q_i}{r + {{\cal Q}}_i - Q_i}
  =\big[ 1- \frac{Q_i}{r} H_i^{-1} \big]^{-1} , \quad
 Q_i =\mu\sinh\delta_i\cosh\delta_i ,
\end{equation}
for the electric (D2-brane) part, and
\begin{equation}
 H_i \to H'_i = 1 + \frac{P_i}{r} , \quad
 P_i =\mu\sinh\gamma_i\cosh\gamma_i\ ,
\end{equation}
for the magnetic (D4-brane) part.
Here $Q_i$ and $P_i$
are the electric and magnetic charges respectively.
The BPS limit is  realized when
$\mu \to 0 ,\  \delta_i\to \infty$, and $\gamma_i\to  \infty$,
while the charges $Q_i$ and $P_i$
are kept fixed.
In this case ${{\cal Q}}_i=Q_i$ and  ${{\cal P}}_i =P_i$, so that  $H'_i=H_i$.
We note that the metric of two intersecting
D-branes are not
extremal in the BPS limit.

(3) In the case when the solution has a null isometry,
i.e. the intersecting D-branes have a common string along
some direction $y$, one can add momentum along
$y$ by applying the coordinate transformation
\begin{equation}
t'= \cosh \sigma \ t - \sinh \sigma \ y , \quad
y'= - \sinh \sigma \ t + \cosh \sigma \ y , \label{sigma}
\end{equation}
to the non-extremal background which is obtained
according to the above two steps. Then we obtain that
\begin{eqnarray*}
 - f(r) dt^2 + dy^2\ &\to& - f(r) dt'^2 + dy'^2 \\
  &=& - dt^2 + dy^2 +
  \frac{\mu}{r} (\cosh \sigma \ dt - \sinh \sigma \ dy)^2,
\end{eqnarray*}
where $\sigma$ is a boost parameter.
This transformation is called the boost transformation.

\section{Surface Gravity and Compactification}
In ten-dimensions~\cite{CT,Tsey,GKT},
the metrics have the compactification radii.
The radii are usually taken to be arbitrary.
However for the generic radii,
we find that the Euler numbers of the metrics are non-integers.
In this section, we consider the surface gravities
of the compactified dimensions.
We constrain the compactification radii
using the surface gravity.
As a result, we obtain the integer Euler numbers.

We consider a way to constrain the radii
in the compactified directions.
We first consider the radius
in the Euclidean time direction.
We treat the Euclidean time coordinate
as the polar angle in the $t-r$ directions.
We define the deficit angle $\delta$
in the $t-r$ directions as,
\begin{eqnarray*}
 \frac{1}{2}\int_0^\beta dt \int_{r_H}^\infty dr
  \sqrt{g_{tt}g_{rr}} R_{(2)} = 2\pi - \delta,
\end{eqnarray*}
where $R_{(2)}$ is the scalar curvature
in the $t-r$ directions.
$\beta$ is the period of the time coordinate.
$r_H$ is the radius of the event horizon,
which satisfies that
 $ g_{tt}\bigg|_{r = r_H} = 0 $.
The metric in the $t-r$ directions is
\begin{eqnarray*}
 ds_{(2)}^2 = g_{tt} dt^2 + g_{rr} dr^2.
\end{eqnarray*}
The scalar curvature $R_{(2)}$ and the extrinsic curvature $K$
are
\begin{eqnarray*}
 R_{(2)} = - \frac{1}{\sqrt{g_{tt}g_{rr}}}
 \partial_r\bigg[\frac{1}{2}\frac{\partial_r g_{tt}}
 {\sqrt{g_{tt}g_{rr}}}\bigg], \quad
 K = \frac{1}{\sqrt{g_{rr}}}
    \bigg[\frac{1}{2}\frac{\partial_rg_{tt}}{g_{tt}}\bigg] .
\end{eqnarray*}
We define the Euler number $\chi$
from the Gauss-Bonnet theorem is
\begin{eqnarray}
 2\pi \chi &=& \frac{1}{2}\int_0^\beta dt
  \int_{r_H}^\infty dr \sqrt{g_{tt}g_{rr}}R_{(2)}
  - \int_0^{\beta} dt \sqrt{g_{tt}}K
     \bigg|_{r=\infty} \nonumber \\
   &=& \frac{\beta}{2}\frac{\partial_r g_{tt}}
 {\sqrt{g_{tt}g_{rr}}} \bigg|_{r=r_H} . \label{chi}
\end{eqnarray}
If no singular effect of the event horizon exists,
then the topology
in $t-r$ directions is ${\bf R^2}$,
then the Euler number is 1.
Therefore, from (\ref{chi}) the radius $\beta$
in the time direction is
\begin{eqnarray*}
\beta = \frac{2\pi}{\kappa_t}, \quad
  \kappa_t &\equiv& \frac{1}{2}\frac{\partial _r g_{tt}}
               {\sqrt{g_{rr}g_{tt}}}\bigg|_{r = r_H},
\end{eqnarray*}
where $\kappa$ is the surface gravity.
$r_H$ is a radius of the black hole event horizon.
We obtain that the deficit angle is $0$,
and the Euler number is 1.
Then if we use the radius $\beta$
in the time direction,
the singular effect of the event horizon $r_H$ is absent.
Therefore we take this radius
in the time direction
to obtain the smooth geometry
around the event horizon.

For example, we find that the Euler numbers
of the Schwarzschild metric, the Kerr-Newman metric,
and the $U(1)$ dilaton metric to be integers
by using the relation between
the surface gravity and the compactification radius
(the inverse temperature).

We extend this idea to the ten-dimensional manifolds
in order to obtain the integer Euler numbers.
We next consider the compactification radii
of the black holes in ten-dimensions.
We treat the black holes which have six compactified
directions.
These compactified directions
are
the same as the Euclidean time direction,
then the topology is ${\bf R^8} \times {\bf S^2}$.
The metric $g_{ii}$ of the compactified directions
does not depend the variables
of the compactified directions.
Therefore we can choose
one of the compactified directions
and consider the Euler numbers
in the $i-r$ directions partly.

We consider the compactification radii in the $i-r$
directions,
where $i$ is one of the compactified directions.
The topologies of the $i-r$ directions are ${\bf R^2}$.
We define the deficit angle $\delta_i$
in the $i-r$ directions as
\begin{eqnarray*}
 \frac{1}{2} \int_0^{\beta_i} dx^i \int_{r_H}^\infty dr
  \sqrt{g_{ii}g_{rr}} R_{(2)} = 2\pi - \delta_i,
\end{eqnarray*}
where $R_{(2)}$ is the scalar curvature
in the $i-r$ directions.
$\beta_i$ are the periods of the $i$-th coordinate.
The metric in the $i-r$ directions is
\begin{eqnarray*}
 ds_{(2)}^2 = g_{ii} dt^2 + g_{rr} dr^2.
\end{eqnarray*}
The scalar curvature $R_{(2)}$ and the extrinsic curvature $K$
are
\begin{eqnarray*}
 R_{(2)} = - \frac{1}{\sqrt{g_{ii}g_{rr}}}
 \partial_r\bigg[\frac{1}{2}\frac{\partial_r g_{ii}}
 {\sqrt{g_{ii}g_{rr}}}\bigg], \quad
 K = \frac{1}{\sqrt{g_{rr}}}
    \bigg[\frac{1}{2}\frac{\partial_rg_{ii}}{g_{ii}}\bigg] .
\end{eqnarray*}
We define the Euler number $\chi$
from the Gauss-Bonnet theorem as
\begin{eqnarray}
 2\pi \chi &=& \frac{1}{2}\int_0^{\beta_i} dx^i
  \int_{r_H}^\infty dr \sqrt{g_{ii}g_{rr}}R_{(2)}
  - \int_0^{\beta_i} dx^i \sqrt{g_{ii}}K
     \bigg|_{r=\infty} \nonumber \\
 &=& \frac{\beta}{2}\frac{\partial_r g_{ii}}
 {\sqrt{g_{ii}g_{rr}}} \bigg|_{r = r_H}  . \label{chi2}
\end{eqnarray}
If no singular effect of the event horizon exists,
then the topology
in $i-r$ directions is ${\bf R^2}$,
and the Euler number is 1.
Therefore, from (\ref{chi2})
we suppose that the compactification radius
$\beta_i$ in the $i$-th direction is defined
by the following "surface gravity $\kappa_i$":
\begin{eqnarray*}
  \beta_i(r_H) &=& \frac{2\pi}{\kappa_i}, \quad
  \kappa_{i}(r_H)
   \equiv \frac{1}{2}\frac{\partial _r g_{ii}}
    {\sqrt{g_{rr}g_{ii}}}\bigg|_{r = r_H}, \\
  && (i=1 \cdots 6) .
\end{eqnarray*}
We obtain that the vanishing deficit angles
in the compactified directions.
Therefore we use this radius $\beta_i$
in the compactified directions
to
obtain the smooth geometry
around the event horizon.

We obtain the integer Euler numbers
for the ten-dimensional manifolds
using these compactification radii.
For example, let us consider the following metric:
\begin{eqnarray*}
 (4,4) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
      \times & \times & \times & \times & \times &  - &  - & -  & -  & - &  \\
      \times & \times & \times &  - &  - & \times & \times & -  & -  & - & .
                 \end{array} \right. $} }
\end{eqnarray*}
In the non-extremal version of this metric, the event horizon is
located at
\begin{eqnarray*}
 r_H = \mu .
\end{eqnarray*}
In the third direction, the compactification radius is
\begin{eqnarray*}
 \beta_{3}(r_H)
 = 4 \pi (\mu + {\cal M}_1)(\mu + {\cal M}_2)^{3/2}/({\cal M}_1-{\cal M}_2)\sqrt{\mu} . \\
\end{eqnarray*}
Using this definition, we indeed find that the Euler number
of this example to be an integer as follows:
\begin{eqnarray*}
 \chi &=& \frac{1}{(4\pi)^{10} 10!} \int dx^{10}
 \epsilon^{abcdefghij}R_{ab}R_{cd}R_{ef}R_{gh}R_{ij} \\
      &=& \frac{1}{(4\pi)^{10}}
          \int_{r_H}^{\infty} dr \int_0^{\beta_t(r)} dt
          \int_0^{2\pi} d\phi \int_0^{\pi} d\theta \sin \theta \\
      &&  \times \prod_{i=1}^{6} \int_0^{\beta_i(r)} dx_i
           R_{tr}R_{\theta\phi}R_{12}R_{34}R_{56} \\
      &=&  2 .
\end{eqnarray*}
We further obtain the same Euler number for all the other metrics
which are related to this metric by T-duality as it will be explained in the
next section.
We emphasize again that the Euler numbers are non-integers
if we consider the compactification radii as arbitrary constants.

\section{Black Hole Entropies and T-duality}
In this section, we study the entropies
of the intersecting D-branes
using the compactification radii
as defined in the previous section.
The entropies are calculated semiclassically by using $S=A/4G$
in this section.
Here $A$ is the area of the event horizon,
and $G$ is the Newton's constant.
We obtain the entropies which are T-duality invariant.
In the BPS limits ($\mu \to 0 $),
we obtain the finite and non-vanishing entropies
only for two intersecting D-branes.
They are found to be proportional to the product
of the charges of each D-brane.
 We further study the entropies of boosted metrics.
We obtain the relations
between the entropies and the internal momenta of
the intersecting D-branes. These relations are the same as
that of the microscopic D-brane picture~\cite{SV,HS,MS}.

We first study the entropies of the metric with the boost parameter
$\sigma = 0$
in (\ref{sigma}).
We define the proper length in the $i$-th direction as:
\begin{eqnarray*}
  L_i(r) &\equiv&
\bigg|\int_{0}^{\beta_{i}(r)}\sqrt{g_{ii}}dx_i \bigg|
 = 4\pi \bigg| \frac{\sqrt{g_{rr}}}{\partial_r (\ln g_{ii})}
\bigg| , \\
  && ( i= 1,\cdots, 6 ) .
\end{eqnarray*}
Using these definitions, the entropies of the black holes are
\begin{eqnarray}
 S = A_8/4G_{10} \bigg|_{r=\mu}
   = L_1L_2L_3L_4L_5L_6A_{\theta\phi}/4G_{10} \bigg|_{r=\mu}.
\end{eqnarray}
Here $A_8$ is the area of the event horizon,
$A_{\theta\phi}$ is
the area in the $\theta-\phi$ directions.

For example, we consider the following metric :
\begin{eqnarray*}
 (4,4) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
     \times & \times & \times & \times & \times &  - &  - & -  & -  & - & \\
     \times & \times & \times &  - &  - & \times & \times & -  & -  & - & .
                 \end{array} \right. $} }
\end{eqnarray*}
In the third direction, the proper length is
\begin{eqnarray*}
 L_3 = \int_{0}^{\beta_{3}}\sqrt{g_{33}}dx_3
     = 4 \pi (r + {\cal M} _1)^{5/4} (r + {\cal M} _2)^{5/4}
     /({\cal M} _1-{\cal M} _2)\sqrt{r},
\end{eqnarray*}
where ${\cal M}_i$ are ${\cal Q}_i$ or ${\cal P}_i$.
With this definition, we find the following relations :
\begin{eqnarray}
 L_3 = L_4 = L_5 = L_6
 &=& 4\pi(r + {\cal M}_1)^{5/4} (r + {\cal M}_2)^{5/4}/({\cal M}_1-{\cal M}_2)\sqrt{r}
 \nonumber \\
 &\equiv& L_a,
 \nonumber \\
 L_1 = L_2
 &=& 4 \pi (r + {\cal M}_1)^{1/4} (r + {\cal M}_2)^{1/4}\sqrt{r}
 \nonumber \\
 &\equiv& L_b. \label{pre}
\end{eqnarray}
Let us consider the proper lengths of the metrics
which are dual to each other.
If we apply the T-duality transformation to
the fourth-direction of (4,4) type metric,
we obtain the (3,5) type metric,
\begin{eqnarray*}
 (3,5) &=& \mbox{ {\scriptsize
        $\left\{ \begin{array}{c|cccccccccc}
     \times & \times & \times & \times & -  &  - &  - & -  & -  & - & \\
     \times & \times & \times &  - & \times & \times & \times & -  & -  & - & .
        \end{array} \right. $} }
\end{eqnarray*}
For this metric, we also obtain the proper lengths
which are the same with (\ref{pre}).
Then the entropies of (3,5) type metrics
are the same with (4,4) type metrics.
We also obtain the identical proper lengths
for all the other metrics
which are dual to each other.
Therefore the entropies of the T-dual metrics are the same.

We can also obtain the analogous relations with the different
numbers of the D-branes.
The area in the $\theta-\phi$ directions is
\begin{eqnarray*}
 A_{\theta\phi}
 = \int_{0}^{\pi} d\theta \int_{0}^{2\pi} d\phi \sin\theta
   \sqrt{g_{\theta\theta}g_{\phi\phi}}
 = 4 \pi (r + {\cal M}_1)^{1/2} (r + {\cal M}_2)^{1/2}r.
\end{eqnarray*}
Then we find that the entropy of two intersecting D-branes is given by
\begin{eqnarray*}
S &=& A_8/4G_{10} \bigg|_{r=\mu}
   = L_a^4L_b^2A_{\theta\phi}/4 \bigg|_{r=\mu} \\
  &=& (4\pi)^7(\mu+{\cal M}_1)^8(\mu+{\cal M}_2)^8 \\
  && /\bigg[({\cal M}_1-{\cal M}_2)^4[\mu({\cal M}_1+{\cal M}_2)+2{\cal M}_1{\cal M}_2]^2 \bigg] ,
\end{eqnarray*}
where $G_{10}$ is the ten-dimensional Newton's constant.
For three intersecting D-branes, the entropy is  given by
\begin{eqnarray*}
S &=& A_8/4G_{10} \bigg|_{r=\mu} \\
   &=& (4\pi)^7((\mu+{\cal M}_1)^8(\mu+{\cal M}_2)^8(\mu+{\cal M}_3)^8\mu^2 \\
    && /\bigg[[\mu^2({\cal M}_1+{\cal M}_2-{\cal M}_3)
        + 2\mu {\cal M}_1{\cal M}_2 +{\cal M}_1{\cal M}_2{\cal M}_3] \\
    && \times [\mu^2({\cal M}_2+{\cal M}_3-{\cal M}_1)
        + 2\mu {\cal M}_2{\cal M}_3 +{\cal M}_1{\cal M}_2{\cal M}_3] \\
    && \times [\mu^2({\cal M}_3+{\cal M}_1-{\cal M}_2)
        + 2\mu {\cal M}_3{\cal M}_1 +{\cal M}_1{\cal M}_2{\cal M}_3] \bigg]^2 .
\end{eqnarray*}
For four intersecting D-branes, the entropy is
\begin{eqnarray*}
S &=& A_8/4G_{10} \bigg|_{r=\mu} \\
  &=& (4\pi)^7((\mu+{\cal M}_1)^8(\mu+{\cal M}_2)^8(\mu+{\cal M}_3)^8(\mu+{\cal M}_4)^8 \\
   && /\mu^6\bigg[
        [\mu^2({\cal M}_1+{\cal M}_2-{\cal M}_3-{\cal M}_4) +2\mu({\cal M}_1{\cal M}_2-{\cal M}_3{\cal M}_4) \\
   &&   \ + ({\cal M}_1+{\cal M}_2){\cal M}_3{\cal M}_4-({\cal M}_3+{\cal M}_4){\cal M}_1{\cal M}_2] \\
   &&   \times
        [\mu^2({\cal M}_1+{\cal M}_3-{\cal M}_2-{\cal M}_4) +2\mu({\cal M}_1{\cal M}_3-{\cal M}_2{\cal M}_4) \\
   &&   \ + ({\cal M}_1+{\cal M}_3){\cal M}_2{\cal M}_4-({\cal M}_2+{\cal M}_4){\cal M}_1{\cal M}_3] \\
   &&   \times
        [\mu^2({\cal M}_1+{\cal M}_4-{\cal M}_2-{\cal M}_3) +2\mu({\cal M}_1{\cal M}_4-{\cal M}_2{\cal M}_3) \\
   &&   \ + ({\cal M}_1+{\cal M}_4){\cal M}_2{\cal M}_3-({\cal M}_2+{\cal M}_3){\cal M}_1{\cal M}_4] \bigg] ^2 . \\
\end{eqnarray*}
We also list the entropy formula for single brane:
\begin{eqnarray*}
 S = A_8/4G_{10} \bigg|_{r=\mu} = (4\pi)^7\mu^6(\mu+{\cal M}_1)^8/{\cal M}_1^6 .
\end{eqnarray*}

If we consider the BPS limit (namely $\mu \to 0$),
we can obtain the finite and non-vanishing entropies
only for the metrics of two intersecting D-branes.
In order to interpret these entropies,
we define the quantized D-brane charges
in the wrapping directions
except for the intersecting directions.
They are the integer numbers.
In the case of $(4,4)$ case, we define the
quantized  magnetic charges of the D4-branes as
\begin{eqnarray*}
 N_1 &=& \frac{L_a^2}{4\pi} \int {\cal F}_{\theta\phi12}
     d\Omega
 \bigg|_{r=1} \\
     &=& (4\pi)^2{\cal M}_1({\cal M}_1{\cal M}_2)^{5/2}/({\cal M}_1-{\cal M}_2)^2, \\
 N_2 &=& \frac{L_a^2}{4\pi} \int {\cal F}_{\theta\phi12}
     d\Omega
 \bigg|_{r=1} \\
    &=& (4\pi)^2{\cal M}_2({\cal M}_1{\cal M}_2)^{5/2}/({\cal M}_1-{\cal M}_2)^2 .
\end{eqnarray*}
Using these charges, we find the entropies as
\begin{eqnarray}
 S = (4\pi)^3 N_1 N_2 /4 . \label{44}
\end{eqnarray}
In the T-dual cases of this metric, (3,5) and (2,6), we define
\begin{eqnarray*}
 N = \left\{
 \begin{array}{ll}
  \frac{L_a^2}{4\pi}\int *{\cal F} & \mbox{(electric)}, \\
  \frac{L_a^2}{4\pi}\int {\cal F}  & \mbox{(magnetic)},
 \end{array} \right.
\end{eqnarray*}
where $\ast$ is Hodge dual in four-dimensions.
Then we find that the entropies for all metrics
of two intersecting D-branes as the same with (\ref{44}).
Therefore we observe that the entropies of
two intersecting D-branes
are the product of the quantized
charges of each D-brane.

Next we consider the case of the large boost parameter
($\sigma \gg 1$).
In this case, the metric becomes
\begin{eqnarray*}
 - f(r) dt^2 + dy^2\ &\to& - f(r) dt'^2 + dy'^2 \\
   &=& - dt^2 + dy^2
   + \frac{\mu}{r} (\cosh \sigma \ dt - \sinh \sigma \ dy)^2 \\
   &\sim& - dt^2 + dy^2
       + \frac{\mu\cosh^2 \sigma}{r} (dt-dy)^2 .
\end{eqnarray*}
Here we have introduced $\mu' = \mu \cosh^2 \sigma$.
Then the event horizon is at $r_H= \mu'$.
The proper length in the $y$- direction is
\begin{eqnarray*}
 L_y = 4\pi[({\cal M}_1+r)({\cal M}_2+r)]^{5/4} \sqrt{r}
     /(2r^2-2{\cal M}_1{\cal M}_2) .
\end{eqnarray*}
For $(3,3), (2,4)$ and $(1,5)$ cases, we find the following entropy
formula as
\begin{eqnarray*}
 S = L_a^4L_bL_yA_{\theta\phi}/4\bigg|_{r=\mu'}.
\end{eqnarray*}
In the $(4,4), (3,5)$ and  $(2,6)$ cases, we have two isometric directions.
If we consider the case that only one of these directions is
boosted, the entropy is given by
\begin{eqnarray*}
 S = L_a^4L_bL_yA_{\theta\phi}/4\bigg|_{r=\mu'}.
\end{eqnarray*}
We may further consider the case that
the metric is boosted in two different directions.
Let us assume that the $y_2$ direction is boosted first
and then the $y_1$ direction is boosted.
After such a process, we obtain the following metric as
\begin{eqnarray*}
 - f(r) dt^2 + dy_1^2 + dy_2^2 \
   &\to& - f(r) dt'^2 + dy_1'^2 + dy_2'^2 \\
   &=& - dt^2 + dy_1^2 + dy_2^2 \\
    && + \frac{\mu}{r}
   (\cosh^2 \sigma_1\cosh^2 \sigma_2\ dt
    - \sinh \sigma_1\ dy_1
    - \cosh^2 \sigma_1\sinh^2 \sigma_2 dy_2)^2 \\
   &\sim& - dt^2 + dy_1^2 +dy_2^2
    + \frac{\mu\cosh^2 \sigma_1 \cosh^2 \sigma_2}{r}
    (dt-dy_2)^2.
\end{eqnarray*}
Therefore we obtain the same metric
with the boost in the single direction.
As a result, we find that the entropies
of the boosted metrics are the same.

According to the definitions in~\cite{SV,HS,MS},
we define the quantized internal momentum $p$
and the charge $n$ in the direction of $y$.
Here $n$ is a number of the state level of the effective
conformal field theory in the D-brane picture.
The momentum $p$ is quantized in term of
the surface area of the event horizon $A_8$.
We define the charge and the momentum as
\begin{eqnarray*}
   n &=& L_y p ,\\
   n &=& \int dA_8 \mu'/r \bigg|_{r=\mu'}\\
     &=& L_a^4L_bL_yA_{\theta\phi} \\
     &\sim& ({\cal M}_1{\cal M}_2)^6 /({\cal M}_1-{\cal M}_2)^4.
\end{eqnarray*}
Using these quantities, we obtain the entropies in the BPS limit
with $\sigma \gg 1$ as follows,
\begin{eqnarray*}
 S \sim \sqrt{N_1N_2n}.
\end{eqnarray*}
This entropy formula of the boosted metrics is the same with
that of the microscopic
D-brane picture~\cite{SV,HS,MS}.

\section{Conclusion}
We have studied the Euler numbers and the entropies
of the non-extremal black holes
which are constructed
by the intersecting D-branes in ten-dimensions.
The Euler numbers are generally integers.
In ten-dimensions, the metrics have
the arbitrary compactification radii.
However for the generic radii,
the Euler numbers of the metrics are non-integers.
To avoid these difficulties,
we need the way to constrain
the compactification radii.

The another physical reasoning that we need to fix
the compactification radii
is in order to avoid the singular effects
of the horizon in the compactified directions.
We have discussed this point in the section 3 .
We consider the way to define the period in the time  directions.
If no singular effects of the horizon exist in $t-r$ directions,
then the topology of these directions are ${\bf R^2}$,
and the Euler number of these directions is 2.
Using the Gauss-Bonnet theorem,
we find that
the Euclidean time coordinate
have the period which is
the inverse of the surface gravity.
Therefore we need to take this period
in the time direction
to avoid the singular effects of the horizon.
Similarly, we need to fix
the compactification radii to avoid the singular effects
of the horizon in the compactified directions.
We find the necessity that
the compactification radii are the inverses of
the surface gravity in the compactified directions
using the Gauss-Bonnet theorem.

We identify the inverse of the surface gravities as
the compactification radii
in order to obtain the integer Euler numbers.
The entropies of the black holes are
T-duality invariant when we use
the compactification radii
which are defined by surface gravities.
In the BPS limit, we have the finite and non-zero entropies
only with two intersecting D-branes.
We have introduced the quantized D-brane
charges $N_1$ and $N_2$.
We find the common relation
of these entropies as $S \sim N_1N_2$,
when the boost parameter vanishes.
Therefore we observe that the entropies
are proportional to the product of the charges of each D-brane.

In the case of the large boost parameter,
we also obtain the entropy formula for
the black holes constructed by two intersecting D-branes.
The entropies of the  boosted black holes
are also T-duality invariant.
We have introduced the quantum number $n$.
Here $n$ is a number of the state level of the
effective conformal field theory in the D-brane picture.
The entropies are $S \sim \sqrt{N_1N_2n}$.
The entropies of the boosted black holes can be interpreted
as the entropies of the microscopic states
in the D-brane picture.

We need to consider the microscopic interpretation of
the entropies in association with the non-boosted
black holes in the D-brane picture.
The entropies of the non-boosted black holes reduce
to $S \sim N_1N_2$.
The black holes have no momenta $n$ in this limit.
On the other hand,
the entropies in the microscopic
D-brane picture $S \sim \sqrt{N_1N_2n}$
are obtained for the large momenta.
Therefore the entropy formula for the non-boosted black holes
could be entirely different from that for the boosted black holes
which agrees with
the microscopic D-brane picture.
The microscopic entropies in the D-brane picture
with small or zero momenta
were not discussed so far.
We further need to study the microscopic
interpretation of the entropies
with the momenta $n \sim 0$ in the D-brane picture.

We have proposed a method to constrain the compactification
radii of the non-extremal black holes of
the intersecting D-branes.
We have identified the inverse of the surface gravities as
the compactification radii.
We have correctly obtained
the integer Euler numbers of the black holes.
Although we do not know another way to constrain
the compactification radii,
we have not shown that the way
to constrain the compactification radii
is unique.
Therefore another way to constrain
the compactification radii might be found.
We further need to study the way
and the physical reasoning
to constrain the  compactification radii.

\acknowledgements
We thank Y. Kitazawa for discussions and
for carefully reading the manuscript
and suggesting various improvements.


\section{Introduction}

In a cold dark matter (CDM) universe, dark matter haloes form hierarchically through accretion and merging(for a recent review, see \citet{Frenk2012}).
Many rigorous and reliable predictions for the halo mass function and subhalo mass functions in CDM are provided by numerical simulations\citep[e.g.][]{springel2009, Gao2011, Colin2000, Hellwing2015, Bose2016}.  When the small haloes merge to the larger systems, they become the subhaloes and suffer from environmental effects such as tidal stripping, tidal heating and dynamical friction that tend to remove the mass from them and even disrupt them \citep{ Tormen1998, Taffoni2003, Diemand2007, Hayashi2003, gao2004, springel2009, xie2015} .  At the mean time, the satellite
galaxies that reside in the subhaloes also experience environmental effects. The tidal stripping and ram-pressure can remove the hot
gas halo from satellite galaxies which in turn cuts off their supply of cold gas and quenches star formation\citep{Balogh2000,Kawata2008,McCarthy2008,Wang2007,Guo2011,Wetzel2013}. Satellite galaxies in some cases also experience a mass loss in the cold gas component and stellar component during the interaction with the host haloes \citep{Gunn1972,Abadi1999,Chung2009, Mayer2001, Klimentowski2007, Kang2008,  Chang2013}.

Overall, the subhaloes preferentially lose their dark mass rather than the luminous mass, because the mass distribution of satellite galaxies is much more concentrated than that of the dissipationless dark matter particles. Simulations predict that the mass loss of infalling subhaloes depends inversely on  their halo-centric radius \citep[e.g.][]{Springel2001, DeLucia2004, gao2004, xie2015}. Thus, the halo mass to stellar mass ratio of satellite galaxies should increase as a function of halo-centric radius.  Mapping the mass function of subhaloes from observations can provide important constraints on this galaxy evolution model.

In observations, dark matter distributions are best measured with gravitational lensing. For dark matter subhalos, however, such observations are
challenging due to their relative low mass compared to that of the host dark matter halo.
The presence of subhalos can cause flux-ratio anomalies in multiply imaged lensing systems \citep{Mao1998,Metcalf2001,Mao2004,xu2009,  Nierenberg2014}, perturb the locations, and change the image numbers \citep{kneib1996,Kneib2011}, and
 disturb the surface brightness of extended Einstein ring/arcs  \citep{Koopmans2005,Vegetti2009a,Vegetti2009b,Vegetti2010,Vegetti2012}.
However, due to the limited number of high quality images and the rareness of strong lensing systems, only a few subhalos have been detected
through strong lensing observation. Besides, strong lensing effects can only probe the central regions of dark matter haloes \citep{Kneib2011}.
Therefore, through strong lensing alone, it is difficult to draw a
comprehensive picture of the co-evolution between subhalos and galaxies.

Subhalos can also be detected in individual clusters
through weak gravitational lensing or weak lensing combining strong lensing
 \citep[e.g.][]{Natarajan2007,Natarajan2009, Limousin2005,Limousin2007,Okabe2014}.
In \citet{Natarajan2009},  Hubble Space Telescope images were used to investigate the
 subhalo masses of $L^*$ galaxies in the massive, lensing cluster,
 Cl0024+16 at z = 0.39, and to study the subhalo
  mass as a function of halo-centric radius. \citet{Okabe2014} investigated subhalos
 in the very nearby Coma cluster with imaging from the Subaru telescope. The deep imaging and the large apparent size of the
cluster allowed them to measure the masses of subhalos selected by
shear alone. They found 32 subhalos in the Coma cluster
and measured their mass function. However, this kind of study requires very high quality images of  massive nearby clusters,
making it hard to extend such studies to large numbers of clusters.

A promising alternative way to investigate the satellite-subhalo
relation is through a stacking analysis of galaxy-galaxy lensing with
large surveys. Different methods have been proposed in previous studies \citep[e.g.][]{Yang2006, Li2013,Pastor2011,Shirasaki2015}.
Although the tangential shear generated by a single subhalo is small,
by stacking thousands of satellite galaxies the statistical noise can
be suppressed and the mean projected density
profile around satellite galaxies can be measured. \citet{Li2014} selected
satellite galaxies in the SDSS group catalog from \citep{Yang2007} and measured the weak lensing signal
around these satellites with a lensing source catalog derived from the CFHT Stripe82 Survey \citep{Comparat2013}. This was the first galaxy-galaxy lensing
 measurement of subhalo masses in galaxy groups. However, the uncertainties of the measured subhalo
masses were too large to investigate the satellite-subhalo relation as a function of halo-centric radius.

In this paper, we apply the same method as  \citet{Li2013,Li2014} to measure the galaxy-galaxy lensing signal
for  satellite galaxies in the SDSS redMaPPer cluster catalog \citep{Rykoff2014, Rozo2014}. Unlike the
the group catalog of \citet{Yang2007} which is constructed using SDSS spectroscopic galaxies, the redMaPPer
catalog relies on photometric cluster detections, allowing it to go to higher redshifts.  As a result,
there are more massive clusters in the redMaPPer cluster
catalog. Therefore, we expect to signal-to-noise of the satellite galaxy lensing signals
to be higher, enabling us to derive better constraints on subhalo properties.

The paper is organized as follows. In section \ref{sec:data}, we describe the lens and
source catalogs. In section \ref{sec:model}, we present our lens model. In section \ref{sec:res},
we show our observational results and our best fit lens model.
The discussions and conclusions are presented in section \ref{sec:sum}. Throughout the paper, we adopt a $\Lambda$CDM cosmology with parameters given by the WMAP-7-year data \citep{komatsu2010} with $\Omega_L=0.728$, $\Omega_{M}=0.272$ and $h \equiv H_0/(100 {\rm km s^{-1} Mpc^{-1}}) = 0.73$. In this paper, stellar mass is estimated assuming a \cite{Chabrier2003} IMF.

\section{Observational Data}
\label{sec:data}

\subsection{Lens selection and stellar masses}
\label{sec:lens_select}

We use satellite galaxies in the redMaPPer clusters as lenses. The redMaPPer cluster
 catalog  is extracted from photometric galaxy samples of the SDSS
  Data Release 8 \citep[DR8,][]{Aihara2011} using the red-sequence Matched-filter
   Probabilistic Percolation cluster
 finding algorithm \citep{Rykoff2014}. The redMaPPer algorithm uses the 5-band
$(ugriz)$  magnitudes of galaxies with a magnitude cut $i<21.0$ over a total area of 10,000 deg$^2$
to photometrically detect galaxy clusters.

redMaPPer uses a multi-color richness estimator $\lambda$,
 defined to be the sum of the membership probabilities over all galaxies.
 In this work,  we  use clusters with richness $\lambda>20 $ and photometric redshift $z_{\rm cluster}<0.5$.
  In the overlapping region with the CFHT Stripe-82 Survey,  we have a total of 634 clusters. For each redMaPPer cluster,
 member galaxies are identified according to their
 photometric redshift,  color and their cluster-centric distance.  To reduce the contamination
 induced by fake member galaxies, we only use satellite galaxies with membership probability $P_{\rm mem}>0.8$.
 The redMaPPer cluster finder identifies 5  central galaxy candidates for each cluster, each with an estimate of the probability $P_{\rm cen}$
 that the galaxy in question is the central galaxy of the cluster.
We remove all central galaxy candidates  from our lens sample. For more details  about
redMaPPer cluster catalog, we refer the readers to \citet{Rykoff2014, Rozo2014}.

Stellar masses are estimated for member galaxies in the redMaPPer
catalog using the Bayesian spectral energy distribution (SED) modeling
code {\tt iSEDfit} \citep{Moustakas2013}.  Briefly, {\tt iSEDfit}
determines the posterior probability distribution of the stellar mass
of each object by marginalizing over the star formation history,
stellar metallicity, dust content, and other physical parameters which
influence the observed optical/near-infrared SED.  The input data for
each galaxy includes: the SDSS $ugriz$ {\tt model} fluxes scaled to the
$r$-band {\tt cmodel} flux; the 3.4- and 4.6-$\micron$ ``forced" WISE
\citep{Wright2010} photometry from \citet{Lang2014}; and the
spectroscopic or photometric redshift for each object inferred from
redMaPPer.  We adopt delayed, exponentially declining star formation
histories with random bursts of star formation superposed, the
Flexible Stellar Population Synthesis (FSPS) model predictions of
\citet{Conroy2009, Conroy2010}, and the \citet{Chabrier2003} initial
mass function (IMF) from 0.1-100 $M_{\odot}$.  For reference, adopting
the \citet{Salpeter1955} IMF would yield stellar masses which are on
average 0.25 dex (a factor of 1.8) larger. We apply a
  stellar mass cut of $M_{\rm star}>10^{10} \rm {h^{-1}M_{\odot}}$ to our satellite
  galaxy sample. In Fig.\ref{fig:mz}, we show the $M_{\rm
    star}$--$z_l$ distribution for our lens samples, where $z_l$ is
  the photometric redshift of the satellite galaxy assigned
  by the redMaPPer algorithm.  The low stellar mass satellite galaxies are
   incomplete at higher redshift, but they will not affect the conclusion of this
   paper.

\begin{figure}
\includegraphics[width=0.4\textwidth]{fig1.pdf}
\caption{The $M_{\rm star}$--$z_l$ distribution of lens galaxies, where
$z_l$ is the photometric redshift, and $M_{\rm star}$ is in units of $M_{\odot}$. }
\label{fig:mz}
\end{figure}

\subsection{The Source Catalog}

The source catalog is measured from the Canada--France--Hawaii
Telescope Stripe 82 Survey (CS82),  which is an $i-$band imaging survey
covering the SDSS Stripe82 region. With excellent seeing conditions --- FWHM between 0.4 to 0.8 arcsec ---
the CS82 survey reaches a depth of $i_{\rm AB}\sim24.0$.
The survey contains a total of 173 tiles, 165 of which from CS82 observations and 8 from CFHT-LS Wide \citep{Erben2013}.
The CS82 fields were observed in four dithered observations with 410s
exposure.
The 5$\sigma$ limiting magnitude is $i_{\rm AB}\sim 24.0$ in a 2 arcsec diameter aperture.

Each CFHTLenS science image is supplemented by a mask,
indicating regions within which accurate photometry/shape measurements
of faint sources cannot be performed, e.g. due to extended
haloes from bright stars. According to \citet{Erben2013}, most of the science analysis are safe with
sources with MASK$\le1$. After applying all the necessary masks and removing overlapping regions,
 the effective survey area reduces from 173 deg$^2$ to
129.2 deg$^2$.  We also require that source galaxies to have FITCLASS= 0, where FITCLASS is the flag that
describes the star/galaxy classification.

Source galaxy shapes are measured with the lensfit method \citep{Miller2007,  Miller2013},
closely following the procedure in \citet{ Erben2009,  Erben2013}. The
shear calibration and systematics of the
lensfit pipeline are described in detail in \citet[][]{Heymans2012}. The specific procedures that
are applied to the CS82 imaging are described in  Erben et al. (2015,  in preparation).

Since the CS82 survey only provides the $i$-band images, the CS82 collaboration derived  source photometric redshifts
using the $ugriz$ multi-color data from the SDSS co-add \citep{Annis2014}, which reaches roughly 2
magnitudes deeper than the single epoch SDSS imaging. The photometric redshift (photo-z) of the background
galaxies were estimated using a Bayesian photo-$z$ code \citep[BPZ,][]{Benitez2000, Bundy2015}.
The effective weighted source galaxy number density is 4.5 per arcmin$^{2}$. Detailed systematic tests for this weak lensing
catalog are described in Leauthaud et al. 2015 (in prep).

\subsection{Lensing Signal Computation}

In a galaxy-galaxy lensing analysis, the excess surface mass density
,  $\Delta\Sigma$ is inferred  by measuring the tangential shear,  $\gamma_t(R)$,  where
\begin{equation}\label{eq:ggl}
\Delta\Sigma(R)=\gamma_t(R)\Sigma_{\rm crit}={\overline\Sigma}(<R)-\Sigma(R)\, ,
\end{equation}
where ${\overline\Sigma}(<R)$ is the mean surface mass density
within $R$,  and $\Sigma(R)$ is the average surface density at the projected radius $R$,  and
$\Sigma_{\rm crit}$ is the lensing critical surface density
\begin{equation}
  \Sigma_{\rm crit}=\frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}}\, ,
\end{equation}
where $D_{ls}$ is the angular diameter distance between the lens and the source,
and $D_l$ and $D_s$
are the angular diameter distances from the observer to the lens and
to the source,  respectively.

We select satellite galaxies as lenses and stack
lens-source pairs in physical radial distance $R$ bins from $0.04$ to $1.5 {\rm h^{-1}Mpc}$.
To avoid contamination from foreground galaxies, we remove lens-source pairs
with $z_{s}-z_{l} < 0.1$, where $z_s$ and $z_l$ are lens redshift and source redshift respectively.
We have also tested the robustness of our results by varying the selection criteria for source galaxies.
We find that selecting lens-source pairs with $z_{s}-z_{l} > 0.05$ or $z_{s}-z_{l} > 0.15$ only changes the
final lensing signal by less than 7\%, well below our final errors.

For a given set of lenses,  $\Delta\Sigma(R)$ is estimated using
\begin{equation}
\Delta\Sigma(R)=\frac{\sum_{l}\sum_{s}w_{ls}\gamma_t^{ls}\Sigma_{\rm crit}(z_l,z_s)}{\sum_{l}\sum_{s}w_{ls}}\, ,
\end{equation}
where
\begin{equation}
w_{ls}=w_n\Sigma_{\rm crit}^{-2}(z_l,z_s)\,
\end{equation}
and $w_n$  is a weight factor defined by Eq.\, (8) in \citet{Miller2013}.
$w_n$ is introduced to account for
 the intrinsic distribution of ellipticity and shape measurement uncertainties.

In the lensfit pipeline, a calibration factor for the multiplicative error $m$ is estimated
for each galaxy based on its signal-to-noise ratio, and the size of the galaxy.
Following \citet{Miller2013}, we account for these multiplicative errors
in the stacked lensing by the correction factor
\begin{equation}
1+K(R)=\frac{\sum_{l}\sum_{s} w_{ls}(1+m_s)}{\sum_{l}\sum_{s} w_{ls}}
\end{equation}

The corrected lensing measurement is as:
\begin{equation}
\Delta\Sigma(R)^{\rm corrected}=\frac{\Delta\Sigma(R)}{1+K(R)}
\end{equation}

In this paper, we stack the tangential shears around satellite galaxies binned according to their
projected halo-centric distance $r_p$, and fit the galaxy-galaxy lensing signal to
obtain the subhalo mass of the satellite galaxies. We describe our theoretical lens
models below.

\section{The Lens Model}
\label{sec:model}

The surface density around a lens galaxy  $\Sigma(R)$ can be written as:
\begin{equation}\label{xi_gm}
\Sigma(R)=\int_{0}^{\infty}\rho_{\rm g, m} \left(\sqrt{R^2+\chi^2} \right)\, {\rm d}\chi\, ;
\end{equation}
and the mean surface density within radius  $R$ is
\begin{equation}\label{xi_gm_in}
\Sigma(< R)=\frac{2}{R^2} \int_0^{R} \Sigma(u) \,  u \,  {\rm d}u ,
\end{equation}
where $\rho_{g, m}$ is the density profile around the lens,  and
$\chi$ is the physical distance along the line of sight.
The excess surface density around a satellite galaxy is composed of three
components:
\begin{equation}
\Delta\Sigma(R)=\Delta\Sigma_{\rm sub}(R)+\Delta\Sigma_{\rm host}(R,  r_p)+\Delta\Sigma_{\rm star}(R)\, ,
\end{equation}
where $\Delta\Sigma_{\rm sub}(R)$ is the contribution of the subhalo in which the satellite
galaxy resides,  $\Delta\Sigma_{\rm host}(R,  r_p)$ is the contribution from the host halo of the cluster/group,
where $r_p$ is the  projected distance from the satellite galaxy  to the center of the host halo,
and $\Delta\Sigma_{\rm star}(R)$ is the contribution from
  the stellar component of the satellite galaxy. We neglect the two-halo term, which is the contribution from
  other haloes on the line-of-sight, because this contribution is only
  important
 at $R>3 {\rm h^{-1}Mpc}$ for clusters \citep[see][]{Shan2015}.
  At small scales where the subhalo term dominates, the contribution
  of the 2-halo term is at least an order of magnitude
  smaller than the subhalo term \citep{Li2009}.

\subsection{Host halo contribution}

We assume that  host haloes are centered on the central galaxies of clusters,
with a density profile following the NFW \citep{NFW97} formula:
\begin{equation}
\rho_{\rm host}(r)=\frac{\rho_{\rm 0,host}}{(1+r/r_{\rm s,host})^2(r/r_{\rm s,host})} \,,
\end{equation}
where $r_{\rm s,host}$ is the characteristic scale of the halo and $\rho_{\rm 0,host}$ is a normalization
factor.  Given the mass of a dark matter halo, its profile then only depends on the concentration
parameter $c\equiv r_{\rm 200}/r_{\rm s,host}$, where $ r_{\rm 200}$ is a radius within which the average
density of the halo equals to 200 times the universe critical density, $\rho_{\rm crit}$.
The halo mass $M_{\rm 200}$ is defined as $M_{\rm 200}\equiv 4\pi/3r_{\rm 200}^3(200 \rho_{\rm crit})$.

Various fitting formulae for mass-concentration relations have been derived from N-body numerical simulations
  \citep[e.g.][]{Bullock2001,Zhao2003,Dolag2004,Maccio2007,Zhao2009, Duffy2008, Neto2007}.
 These studies find that the concentration decreases with increasing
 halo mass. Weak lensing observations also measure this trend,
 but the measured amplitude of the mass-concentration relation is  slightly smaller than that in the simulation
 \citep[e.g.][]{Mandelbaum2008, Miyatake2013, Shan2015}.
Since there is almost no degeneracy between  the subhalo mass and the
concentration \citep{Li2013,George2012},
we expect that the exact choice of the mass-concentration relation
should not have a large impact on our conclusions.
 Throughout this paper,  we adopt the mass concentration relation from
 \citet{Neto2007}:
 %
 \begin{equation}
 c=4.67(M_{\rm 200}/10^{14} \rm {h^{-1}M_{\odot}})^{-0.11}
 \end{equation}
 %

We stack satellite galaxies with different halo-centric distance  $r_p$, and in host halos
with different mass. Thus the lensing contribution from the
host halo is an average of $\Delta{\Sigma}_{\rm host}$ over $r_p$ and
host halo mass $M_{\rm 200}$. The host halo profile is modeled as follows.

For each cluster, we can estimate its mass via the richness-mass relation of \citet{Rykoff2012}:
\begin{equation}
\label{eq:mass_rich}
  \ln{\left( \frac{M_{\rm 200}}{h_{\rm 70}^{-1}10^{14}M_{\odot}} \right)} = 1.48 + 1.06\ln(\lambda/60)
\end{equation}
In the redMaPPer catalog,  each cluster has five possible central galaxies, each with probability $P_{\rm cen}$.
We assume that the average $\Delta\Sigma$ contribution from the host halo to a satellite can be written as:
\begin{equation}
  \Delta\bar{\Sigma}_{\rm host}(R)=A_0 \sum_{i}^{N_{\rm sat}}\sum_{j}^5\Delta\Sigma_{\rm host}(R|r_{\rm p, j},  M_{\rm 200}) P_{\rm cen, j}\, ,
\end{equation}
where $r_{\rm p, j}$ is the projected distance between the satellite and the jth candidate of the host cluster center,  $P_{\rm cen, j}$
is the probability of jth candidate to be the true center, and $N_{\rm
  sat}$  is the number of stacked satellite galaxies. $A_0$ is
the only free parameter of the host halo contribution model. It describes an adjustment of the lensing amplitude. If the richness-mass relation
is perfect, the best-fit $A_0$ should be close to unity.  Note that, the subhalo mass determination is robust against the variation of
the normalization in richness-mass relation. If we decrease the normalization in Eq. \ref{eq:mass_rich} by 20\%, the best-fit subhalo mass
changes only by 0.01 dex, which is at least 15 times smaller than the $1\sigma$ uncertainties of $M_{\rm sub}$ (see table \ref{tab:para_nfw}).

\subsection{Satellite contribution}

In  numerical simulations, subhalo density profiles are found to
be truncated in the outskirts \citep[e.g.][]{Hayashi2003,springel2009, gao2004, xie2015}.
In this work, we use a truncated NFW profile \citep[][tNFW, hereafter]{Baltz2009, Oguri2011}
to describe the subhalo mass distribution:
\begin{equation}\label{eq:rhosub}
\rho_{\rm sub}(r)=\frac{\rho_{0}}{r/r_s(1+r/r_s)^2}\left(\frac{r_t^2}{r^2+r_t^2}\right)^2 \,
\end{equation}
where $r_t$ is the truncation radius of the subhalo, $r_s$ is the characteristic radius of the tNFW profile
and $\rho_0$ is the normalization.  The enclosed mass with $x\equiv r/r_s$ can be written as:%

\begin{eqnarray}
M(<x)&=&4\pi\rho_0 r_s^3 \frac{\tau^2}{2(\tau^2+1)^3(1+x)(\tau^2+x^2)}\nonumber\\
&&\hspace*{-16mm}\times\Bigl[(\tau^2+1)x\left\{x(x+1)-\tau^2(x-1)(2+3x)
-2\tau^4\right\}\nonumber\\
&&\hspace*{-16mm}+\tau(x+1)(\tau^2+x^2)\left\{2(3\tau^2-1)
{\rm arctan}(x/\tau)\right.\nonumber\\
&&\hspace*{-16mm}\left.+\tau(\tau^2-3)\ln(\tau^2(1+x)^2/(\tau^2+x^2))\right\} \Bigr],
\label{eq:mbmo_nodim}
\end{eqnarray}
where $\tau\equiv r_t/r_s$. We define the subhalo mass $M_{\rm sub}$ to be the total mass within
$r_t$. Given $M_{\rm sub}$, $r_s$ and $r_t$, the tangential  shear $\gamma_t$ can be derived analytically
(see the appendix in \citet{Oguri2011}).

Previous studies have sometimes used instead the pseudo-isothermal elliptical mass
distribution (PIEMD) model derived by \citet{Kassiola1993}) for modeling the mass distribution
around galaxies \citep[e.g.][]{Limousin2009,Natarajan2009,Kneib2011}.
The surface density of the PIEMD model can be written as:
\begin{equation}\label{eq:PIEMD}
\Sigma(R)=\frac{\Sigma_0 R_0}{1-R_0/R_t}\left(\frac{1}{\sqrt{R_0^2+R^2}} - \frac{1}{\sqrt{R_t^2+R^2}} \right)\,
\end{equation}
where $R_0$ is the  core radius, $R_t$ is the truncation radius, and $\Sigma_0$ is a characteristic surface density.
The subhalo mass $M_{\rm sub}$, which is defined to be the enclosed mass with $R_t$, can be written as:
\begin{equation}
M_{\rm sub}=\frac{2\pi \Sigma_0 R_0 R_t}{R_t-R_0}\left[ \sqrt{R_0^2-R^2} - \sqrt{R_t^2-R^2}+(R_t-R_0)\right]
\end{equation}
In this paper, we will fit the data with both tNFW and PIEMD models.

Finally, the lensing contributed from stellar component is usually modeled as a point mass:
\begin{equation}
\Delta\Sigma_{\rm star}(R)=\frac{ M_{\rm star}}{R^2}\,,
\end{equation}
where $M_{\rm star}$ is the total stellar mass of the galaxy.

\section{Results}
\label{sec:res}
\subsection{Dependence on the projected halo-centric radius}

To derive the subhalo parameters, we calculate the $\chi^2$ defined as:
\begin{equation}
 \chi^{2}=\sum_{ij}({\Delta\Sigma(R_i)-\Delta\Sigma(R_i)^{obs}})(\widehat{C_{ij}^{-1}})({\Delta\Sigma(R_j)-\Delta\Sigma(R_j)^{obs}} )\,,
\label{chi2}
\end{equation}
where $\Delta\Sigma(R_i)$ and $\Delta\Sigma(R_i)^{obs}$ are the model and  the observed lensing signal in the $i$th radial bin.
The matrix $C_{ij}$ is the covariance matrix of data error between different radius bins. Even if the ellipticity of different sources are independent,
the cross term of the covariance matrix could still be non-zero, due to some source galaxies are used more than once\citep[e.g.][]{Han2015}.
The covariance matrix can be reasonably calculated with bootstrap method using the survey data themselves\citep{Mandelbaum2005}.
In the paper, We divide the CS82 survey area into 45 equal area sub-regions. We then generate 3000 bootstrap samples by resampling the 45 sub-regions of CS82 observation data sets and calculate the covariance matrix using the bootstrap sample. Thus, the likelihood function can be given as:
\begin{equation}
 L\propto \exp\left(-\frac{1}{2}\chi^2\right).
\label{likelihood}
\end{equation}

We select satellite galaxies as described in section \ref{sec:lens_select}
and measure the stacked lensing signal around satellites in three
$r_p$ bins: $0.1<r_p<0.3$; $0.3<r_p<0.6$ and $0.6<r_p<0.9$(in unit of ${\rm h^{-1}Mpc}$).

For each bin, we fit the stacked lensing signal with a
Monte-Carlo-Markov-Chain(MCMC) method. For the tNFW subhalo case,
we have 4 free parameters:  $M_{\rm sub}$, the subhalo mass;  $r_s$, the tNFW profile
scale radius; $r_t$, the tNFW truncation radius, and $A_0$, the normalization factor
 of the host halo lensing contribution.

We adopt flat priors with broad boundaries for these model parameters.
We set the upper boundaries for  $r_t$ and $r_s$ to be the value of the viral radius and
the scale radius of an NFW halo of $10^{13}\rm {h^{-1}M_{\odot}}$.  We choose these values because
the subhalo masses of satellite galaxy in clusters are expected to be much smaller than
$10^{13}\rm {h^{-1}M_{\odot}}$  \citep{Gao2012}.

For the PIEMD case, we also have 4 free
parameters: $M_{\rm sub}$, $R_0$, $R_t$ and $A_0$. We set the upper boundary of
 $R_t$ to be the same as $r_t$ in the tNFW case. We set the upper boundary of $R_0$ to
  be 20 kpc/h, which is much higher than the typical size of $R_0$  in
  observations\citep{Limousin2005, Natarajan2009}.  We believe our choice of priors is very conservative.
  The detailed choices of priors are listed in table~\ref{tab:prior}.

\begin{table}
\begin{center}
  \caption{Flat priors for model parameters. $M_{\rm sub}$
  is in units of $\rm {h^{-1}M_{\odot}}$ and distances are in units of ${\rm h^{-1}Mpc}$.}

\begin{tabular}{|c|c|c|}
 \hline
 &&\\
 &lower bound& upper bound \\
   &&\\
  \hline
 $A_{\rm 0}$          & 0.3& 2 \\
  \hline
 $\log{M_{\rm sub}} $ & 9 & 13 \\
  \hline
 $r_t$ (tNFW)         & 0 & 0.35 \\
  \hline
 $r_s$ (tNFW)        & 0& 0.06 \\
  \hline
 $R_0$(PIEMD)        & 0 & 0.02\\
  \hline
 $R_t$(PIEMD)        & 0 & 0.35\\
  \hline
\end{tabular}
\label{tab:prior}
\end{center}
\end{table}

In Fig.\ref{fig:gglensing}, we show the observed galaxy-galaxy lensing
signal.  Red dots with error bars represent the observational
data. The vertical error bars show 1 $\sigma$ errors estimated with
the bootstrap resampling the lens galaxies.
Horizontal error bars show the range of each
radial bin. The lensing signals show clearly the characteristic shape
described in figure 2 of \citet{Li2013}.  The lensing signal from the
subhalo term  dominates the central part. On the other hand, the contribution from
the host halo is nearly zero on small scale, and decreases to negative values at intermediate
scale.   This is because  $\Sigma_{\rm host}(R)$ becomes increasingly larger compared to $\Sigma_{\rm host}(<R)$
at intermediate R.  At radii where $R>r_p$, $\Delta\Sigma_{\rm host}(R)$ increases  with R again. At large
scales where $R>>r_p$, $\Delta\Sigma(R,r_p)$ approaches
$\Delta\Sigma_{\rm host}(R,0)$.

The solid lines show the best-fit results with the tNFW model. Dashed lines of different colors show the contribution from
different components. The best-fit model parameters are listed in
table~\ref{tab:para_nfw}. Note that, the value of the first point in
the $[0.6, 0.9]$ $r_p$ bin is very low, which may be due to the relatively
small number of source galaxies in the inner most radial bin. We exclude
this point when deriving our best fit model. For comparison,
we also show the best-fit parameters including this point
in table~\ref{tab:para_nfw}.

Fig.~\ref{fig:2dcontour} shows an example of the MCMC  posterior
distribution of  the tNFW model parameters for satellites in the $[0.3, 0.6]{\rm h^{-1}Mpc}$ $r_p$ bin.
The constraints on the subhalo mass $\log{M_{\rm sub}}$ are tighter than
that in \citet{Li2014}($\sim\pm 0.7$). The amplitude of $A_0$ is slightly smaller than unity, implying that
the clusters may be less massive than predicted by the mass-richness
relation. However, we are still unable to obtain significant
constraints on the structure parameters of the sub-halos.
 In principle, the density profile cut-off caused by tidal effects
can be measured with tangential shear. However, the galaxy-galaxy lensing
measurement stacks many satellites, leading to smearing of the signal.
With the data used here,  the tidal radius as a free parameter are not constrained.
Some galaxy-galaxy lensing investigations introduced additional
constraints in order to estimate the tidal radius. For example, \citet{Gillis2013b}
assumed that galaxies of the same stellar mass but in different environments
have similar sub-halo density profiles except the cut-off radius.
With this additional assumption, they obtained $r_{\rm tidal}/r_{200}=0.26\pm0.14$.
During the review process of our paper, \citet{Sifon2015} posted
a similar galaxy-galaxy lensing measurement of satellite galaxies using
the KiDs survey, and they also found that their data cannot distinguish
models with or without tidal truncation.

In Fig.\ref{fig:piemd}, we show the fitting results of the PIEMD model.
The best-fit parameters are listed in table~\ref{tab:para_piemd}.
For reference, we also over-plot the theoretical prediction of the best-fit
tNFW model with blue dashed lines. Both models provide a good description of the data. The best-fitted $M_{\rm sub}$
and $A_0$ of the two models agree well with each other, showing the validity
of our results.

In numerical simulations, subhalos that are close to host halo centers
are subject to strong mass stripping\citep{Springel2001, DeLucia2004, gao2004, xie2015}.  The mass loss
fraction of subhalos increases from $\sim 30\%$ at $r_{\rm 200}$ to $~90\%$
at 0.1$r_{\rm 200}$ \citep{gao2004,xie2015}. From galaxy-galaxy lensing in
this work, we also find that the $M_{\rm sub}$ of $[0.6, 0.9]$ $r_p$ bin is
much larger than that in $[0.1, 0.3]$ $r_p$ bin (by a factor of 18).  In
Fig.\ref{fig:ML}, we plot the subhalo mass-to-stellar mass ratio
for three $r_p$ bins. The $M_{\rm sub}/M_{\rm star}$ ratio
in the $[0.6, 0.9]{\rm h^{-1}Mpc}$ $r_p$ bin is about 12 times larger than that of the $[0.1, 0.3]{\rm h^{-1}Mpc}$ bin.
If we include the first point in the $[0.6, 0.9]{\rm h^{-1}Mpc}$ bin, the $M_{\rm sub}/M_{\rm star}$
of the tNFW model decreases by 40\%.

For reference, we over-plot the $M_{\rm sub}/M_{\rm star}$--$r_p$ predicted by the
semi-analytical model of \citet{Guo2011}. We adopt the mock galaxy
catalog generated with the \citet{Guo2011} model using the Millennium
simulation \citep{Springel2006}. We select mock satellite galaxies with
stellar masses $M_{\rm star}>10^{10}\rm {h^{-1}M_{\odot}}$ from clusters with $M_{\rm 200}>10^{14}/\rm {h^{-1}M_{\odot}}$.
The median $M_{\rm sub}/M_{\rm star}$-$r_p$ relation is shown in Fig.\ref{fig:ML} with a
black solid line. The green shaded region represents the parameter space where 68\%
of mock satellites distributes. We see the semi-analytical
model predicts an increasing $M_{\rm sub}/M_{\rm star}$ with $r_p$, but with
a flatter slope than in our observations.
Note that we have not attempted to recreate our detailed observational procedure here,
so source and cluster selection might potentially explain this discrepancy.
Particularly relevant here is the fact that our analysis relies on a probability cut
$P_{\rm mem}>0.8$ for satellite galaxies, which implies that $\sim 10\%$ of our
satellite galaxies may not be true members of the clusters, but galaxies
on the line of sight. This contamination is difficult to eliminate completely in
galaxy-galaxy lensing, because the uncertainties in the line-of-sight distances
are usually larger than the sizes of the clusters themselves. In \citet{Li2013}, we
used mock catalogs constructed from $N$-body  simulations to investigate
the impact of interlopers. It is found that 10\% of the galaxies identified as satellites
are interlopers, and this introduces a contamination of $15\%$ in the lensing signal.
We expect a comparable level of  bias here, as shown in Appendix A.
It should be noted, however, that the average membership probability of
our satellite sample does not  change significantly with $r_p$, implying that
the contamination by fake member galaxies is similar at different $r_p$.
We therefore expect that the contamination by interlopers will not lead to
any qualitative changes in the shapes of the density profiles.
A detailed comparison between the observation and simulation,
taking into account the impact of interlopers,  will be carried out in
a forthcoming paper.

\begin{figure}
\includegraphics[width=0.4\textwidth]{fig2.pdf}
\caption{Observed galaxy-galaxy lensing signal for satellite galaxies as a
function of the radius. The top,  middle and bottom panels show results
for satellites with $r_p$ in the range
$[0.1, 0.3]{\rm h^{-1}Mpc}$,  $[0.3, 0.6]{\rm h^{-1}Mpc}$ and $[0.6, 0.9]{\rm h^{-1}Mpc}$, respectively. Red points with error bars
 represent the observational data. The vertical error bars show the
1 $\sigma$ bootstrap error. The horizontal error bars show the bin size. Black solid lines
show the best-fit tNFW model prediction. Dashed lines of different colors show the contribution
from the subhalo, the host halo, and the stellar mass respectively.}
\label{fig:gglensing}
\end{figure}

\begin{figure*}
\includegraphics[width=\textwidth]{fig3.pdf}
\caption{The contours show 68\% and 95\% confidence intervals for the tNFW model parameters,
  $M_{\rm sub}$,  $r_s$,   $r_t$ and $A_0$. Results are shown for satellites with $r_p=[0.3, 0.6]{\rm h^{-1}Mpc}$.
  The last panel of each row shows the 1D marginalized posterior distributions.  Note that the plotting range
  for $r_s$ and $r_t$ is exactly the same as the limits of our prior, so we actually do not have much constraints
  for these two parameters, except that high values are slightly
  disfavored for both.  }
\label{fig:2dcontour}
\end{figure*}
\begin{figure}
\includegraphics[width=0.4\textwidth]{fig4.pdf}
\caption{This figure is similar to Fig\ref{fig:gglensing}. Solid lines represent the
    best-fit PIEMD model prediction and blue dashed lines are the
    predictions of the
        tNFW model.}
\label{fig:piemd}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig5.pdf}
\caption{The subhalo mass-to-stellar mass ratio for galaxies in different $r_p$ bins. Vertical
error bars show the 68\% confidence interval. Empty circles show the
best-fitted $M_{\rm sub}/M_{\rm star}$ values for $[0.6, 0.9]{\rm h^{-1}Mpc}$ $r_p$ bin including
the inner most observational  point (see the bottom panel in Fig.\ref{fig:gglensing}).
 Horizontal error bars show the bin range of $r_p$. The green shaded region represents the parameter space where 68\% of the
semi-analytical satellite galaxies distributes. The semi-analytical model also predicts an increasing $M_{\rm sub}/M_{\rm star}$ with $r_p$, but with
a flatter slope.}
\label{fig:ML}
\end{figure}
\begin{table*}
\begin{center}
  \caption{The best-fit values of the  tNFW model parameters for the stacked
  satellite lensing signal in different $r_p$ bins.
   $\log{M_{\rm sub}} $, and $A_{\rm 0}$ are the best-fit values for the subhalo mass,
    and the normalization factor. We do not
    show the best-fit of $r_s$ and $r_t$ as they are poorly constrained.
     $N_{\rm sat}$  is the number of satellite galaxies in each bin.  $\langle \log M_{\rm star}\rangle$ is the average
      stellar mass of satellites. All errors indicate the 68\% confidence intervals. Masses are in units of $\rm {h^{-1}M_{\odot}}$.
      In our fiducial fitting process, we exclude the first point in $[0.6,0.9]{\rm h^{-1}Mpc}$ $r_p$ bin as a outlier (see Fig.\ref{fig:gglensing}).
        For comparison, the bottom row of the table shows the fitting results including this first point. }

\begin{tabular}{l|c|c|c|c|c|c}
 \hline
 &&&&\\
  $r_p$ range
& $\langle \log(M_{\rm star}) \rangle$
& $\log{M_{\rm sub}} $
& $A_{\rm 0}$
& $M_{\rm sub}/ M_{\rm star}$
&$N_{\rm sat}$ & $\langle z_l \rangle$\\
   &&&&\\
  \hline
   & & && \\
   $[0.1, 0.3]$ &  10.68 & $ 11.37 ^{+ 0.35 }_{- 0.35}$& $ 0.80^{+ 0.01}_{- 0.01}$ & $ 4.43^{+ 6.63}_{- 2.23}$ & 3963 &0.33 \\
 &  & && \\
 \hline
  &&&&\\
$[0.3, 0.6]$ &  10.72 & $ 11.92 ^{+ 0.19 }_{- 0.18}$& $ 0.86^{+ 0.02}_{- 0.02}$ & $ 17.23^{+ 6.98}_{- 6.84} $ &2507 & 0.29\\
 &  & & &\\
 \hline
  &&&&\\
$[0.6, 0.9]$ &  10.78 & $ 12.64 ^{+ 0.12 }_{- 0.11}$& $ 0.81^{+ 0.04}_{- 0.04}$ & $ 75.40^{+ 19.73}_{- 19.09} $&577 &0.24\\
 &  & & &\\
  \hline
  &&&&\\
$[0.6, 0.9]^*$ &  10.78 & $ 12.49 ^{+ 0.13 }_{- 0.13}$& $ 0.81^{+ 0.04}_{- 0.04}$ & $ 54.64^{+ 15.58}_{- 15.80} $&577 &0.24\\
  &&&&\\
 \hline

\end{tabular}
\label{tab:para_nfw}
\end{center}
\end{table*}

\begin{table}
\begin{center}
  \caption{best-fitted parameters with PIEMD model.  }

\begin{tabular}{l|c|c|c|c}
 \hline
 &&&\\
  $r_p$ range
& $\langle \log(M_{\rm star}) \rangle$
& $\log{M_{\rm sub}} $
& $A_{\rm 0}$
& $M_{\rm sub} /M_{\rm star}$ \\
   &&&\\
  \hline
   & & & \\

$[0.1, 0.3]$ &  10.68 & $ 11.30 ^{+ 0.55 }_{- 0.57}$& $ 0.80^{+ 0.01}_{- 0.01}$ & $ 6.72^{+ 7.84}_{- 5.59} $\\
 &  & & \\
 \hline
  &&&\\
$[0.3, 0.6]$ &  10.72 & $ 11.76 ^{+ 0.33 }_{- 0.32}$& $ 0.86^{+ 0.01}_{- 0.01}$ & $ 14.40^{+ 9.01}_{- 9.16} $\\
 &  & & \\
 \hline
  &&&\\
$[0.6, 0.9]$ &  10.78 & $ 12.80 ^{+ 0.15 }_{- 0.15}$& $ 0.79^{+ 0.04}_{- 0.04}$ & $ 110.98^{+ 35.76}_{- 36.49} $\\
  &&&\\
 \hline
   &&&\\
$[0.6, 0.9]^*$ &  10.78 & $ 12.84 ^{+ 0.13 }_{- 0.13}$& $ 0.79^{+ 0.04}_{- 0.04}$ & $ 119.74^{+ 34.34}_{- 35.47} $\\
   &&&\\
 \hline
\end{tabular}
\label{tab:para_piemd}
\end{center}
\end{table}

\subsection{Dependence on satellite stellar mass}

In the CDM structure formation scenario, satellite galaxies with larger stellar mass tend to occupy larger
haloes at infall time \citep[e.g.][]{Vale2004, Conroy2006,Yang_etal2012,Lu_etal2014}.
In addition, massive haloes may retain more mass than lower mass ones
at the same halo-centric radius \citep[e.g.][]{Conroy2006,Moster2010,Simha2012}.
For these two reasons,  we expect that satellite galaxies with larger stellar
mass reside in more massive subhalos.

To test this prediction, we select all galaxies with $r_p$ in the range $[0.3, 0.9]$ ${\rm h^{-1}Mpc}$,  and split
the them into two subsamples: $10<\log(M_{\rm star}/\rm {h^{-1}M_{\odot}})<10.5$ and $11<\log(M_{\rm star}/\rm {h^{-1}M_{\odot}})<12$.
The galaxy-galaxy lensing signals of the two satellite samples are shown in Fig.\ref{fig:mstar}. It is clear
that at small scales, where subhalos dominate, the lensing signals are larger around the more massive
satellites. The best-fit subhalo  mass for the high mass and low mass satellites are
$\log(M_{\rm sub}/\rm {h^{-1}M_{\odot}})=11.14 ^{+ 0.66 }_{- 0.73}$ ($M_{\rm sub}/M_{\rm star}=19.5^{+19.8}_{-17.9}$) and
$\log(M_{\rm sub}/\rm {h^{-1}M_{\odot}})=12.38 ^{+ 0.16 }_{- 0.16}$ ($M_{\rm sub}/M_{\rm star}=21.1^{+7.4}_{-7.7}$ respectively.

\begin{figure}
\includegraphics[width=0.5\textwidth]{fig6.pdf}
\caption{Observed galaxy-galaxy lensing signal  for  satellite
  galaxies in different stellar mass bins at fixed $r_p$. The
legend shows the $\log(M_{\rm star}/\rm {h^{-1}M_{\odot}})$ range for different data points.
The solid lines are the best-fit tNFW models.}
\label{fig:mstar}
\end{figure}

\section{Discussion and conclusion}
\label{sec:sum}

In this paper, we present measurements of the galaxy-galaxy lensing signal of satellite galaxies in redMaPPer
clusters. We select satellite galaxies from massive clusters (richness $\lambda>20$) in the redMaPPer catalog.
We fit the galaxy-galaxy lensing signal around the satellites using a tNFW profile and a PIEMD profile, and obtain the  subhalo masses.
We bin satellite galaxies according to their projected halo-centric distance $r_p$
and find that the best-fit subhalo mass of satellite galaxies increases with $r_p$. The best-fit $\log({M_{\rm sub}/\rm {h^{-1}M_{\odot}}})$
for satellites in the $r_p=[0.6, 0.9]{\rm h^{-1}Mpc}$ bin is larger than in the $r_p=[0.1, 0.3]{\rm h^{-1}Mpc}$ bin by a factor of 18.
The $M_{\rm sub}/M_{\rm star}$ ratio is also found to increase as a function of $r_p$ by a factor of 12. We also find that satellites with more stellar
mass tend to populate more massive subhalos. Our results provide evidence for the tidal striping effects on the subhalos of red sequence satellite galaxies,
as expected on the basis of the CDM hierarchical structure formation scenario.

Many previous studies have tried to test this theoretical prediction
using gravitational lensing. Most of these previous studies focus on measuring
subhalo mass in individual rich clusters. For example, \citet{Natarajan2009} report
the measurement of dark matter subhalos as a function of projected
halo-centric radius for the cluster C10024+16. They found that the mass of
dark matter subhalos hosting early type $L^{*}$ galaxies increases
by a factor of 6 from a halo-centric radius $r<0.6$ ${\rm h^{-1}Mpc}$ to $r\sim 4$
${\rm h^{-1}Mpc}$.  In recent work, \citet{Okabe2014} present the weak
lensing measurement of 32 subhalos in the very nearby Coma cluster.  They
also found that the mass-to-light ratio of subhalos increase as a
function of halo-centric radius: the $M/L_{\rm i'}$ of subhalos
increases from 60 at 10$'$ to about 500 at 70$'$.

Our work is complementary to these results. In the paper,
we measure the galaxy-galaxy lensing signal of subhalos in a statistical sample of rich
clusters. Our results lead to similar conclusions and show evidence
for the effects of tidal striping.  However, due to the statistical noise of the current survey,
there is still a large uncertainties in the measurement of $M_{\rm
  sub}/M_{\rm star}$.

Theoretically, the galaxy - halo connection can be established
by studying how galaxies of different properties reside in dark matter
halos of different masses, through models  such as the halo occupation
models \citet[e.g.][]{Jing_etal1998, PeacockSmith2000,
BerlindWeinberg2002}, the conditional luminosity functions models
\citep[e.g.][]{Yang_etal2003}, (sub)halo-abundance-matching
\citep[ e.g.][]{Vale2004, Conroy2006, Wang2006, Yang_etal2012,
Chaves-Montero2015}, and empirical models of star formation and mass assembly
of galaxies in dark matter halos \citep[e.g.][]{Lu_etal2014, Lu_etal2015}.
In most of these models, the connection is made between galaxy luminosities
(stellar masses) and halo masses before a galaxy becomes a satellite, i.e.
before a galaxy and its halo have experienced halo-specific
environmental effects. The tidal stripping effects after a halo becomes
a sub-halo can be followed in dark matter simulations.
This, together with the halo-galaxy connection established through
the various models,  can be used to predict the subhalo - galaxy
relation at the present day. During the revision of this paper,
\citet{Han2016} posted a theoretical work on subhalo spatial and mass
distributions. In Sec. 6.4 of their paper, they presented how our lensing
measurements can be derived theoretically from a subhalo abundance
matching point of view.  With future data, the galaxy-galaxy lensing
measurements of subhalos associated with satellites are expected
to provide important constraints on galaxy formation and evolution in
dark matter halos.

The results of this paper also demonstrate the promise of next generation weak
lensing surveys. In \citet{Li2013}, it is shown that the constraints on subhalo
structures and $M_{\rm sub}/M_{\rm star}$ can be improved dramatically with
next generation galaxy surveys such as LSST, due to the increase in both sky
area (17000 deg$^2$) and the depth (40 gal/arcmin$^2$), which is crucial
in constraining the co-evolution of satellite galaxies and subhalos.
The space mission, such as Euclid, will survey a similar area of the
sky (20000 deg$^2$) but with much better image resolutions
(FWHM $0.1"$ versus $0.7"$  for LSST), which is expected to
provide even better measurements of galaxy-galaxy lensing. The method
described  here can readily be extended to these future surveys.

\section*{Acknowledgements}

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA,
at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research
Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la
Recherche Scientifique (CNRS) of France, and the University of Hawaii. The Brazilian
partnership on CFHT is managed by the Laborat\'orio Nacional de Astrof\`isica (LNA).
This work made use of the CHE cluster, managed and funded by ICRA/CBPF/MCTI,
with financial support from FINEP and FAPERJ. We thank the support of the
Laboratrio Interinstitucional de e-Astronomia (LIneA). We thank the CFHTLenS
team for their pipeline development and verification upon which much of this surveys pipeline was built.

LR acknowledges the NSFC(grant No.11303033,11133003), the support from
Youth Innovation Promotion Association of CAS. HYS acknowledges the support from
Marie-Curie International Fellowship (FP7-PEOPLE-2012-IIF/327561), Swiss National
Science Foundation (SNSF) and NSFC of China under grants 11103011.
HJM acknowledges support of NSF AST-0908334, NSF AST-1109354 and NSF AST-1517528.
JPK acknowledges support from the ERC advanced grant LIDA and from CNRS.
TE is supported by the Deutsche Forschungs-gemeinschaft through the
Transregional Collaborative Research Centre TR 33 - The Dark Universe.
AL is supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
BM acknowledges financial support from the CAPES Foundation grant 12174-13-0.
MM is partially supported by CNPq (grant 486586/2013-8) and FAPERJ (grant E-26/110.516/2-2012).