Eur. Phys. J. C (2019) 79 :602 Page 3 of 11 602
3 Conditions for a physically realistic core-envelope
model
In order to make the model physically viable, one needs to
verify the following conditions in core, envelope and exte-
rior regions (conditions developed by [47] are further aug-
mented):
(i) Geometrical regularity: The metric potentials and mat-
ter variables should be defined at the center and should
be well behaved throughout within the star [47].
(ii) Density and pressures trends: The matter density ρ,
radial pressure p
r
and transverse pressure p
t
the core
and envelope of the star should be continuous at the
junction, positive and monotonically decreasing out-
ward. Further, the pressure-density ratios should be
positive and less than 1 throughout within the star (Zel-
dovich’s condition [51]) and continuous at the junc-
tion.
(iii) Mass-radius relation, Red-shift and Compactifica-
tion factor: The mass function m(r), compactification
parameter u(r) and gravitational red shift z(r) for the
core and the envelope of the star should be continuous
at the interface and increasing and decreasing respec-
tively with the radial coordinate r.
(iv) Anisotropic constant: The radial pressure should coin-
cides with the tangential pressure at the core of the star
and should be continuous at the interface besides being
asymptotic and increasing outward.
(v) Causal nature of radial sound speed: The radial sound
speed of a compact star model should satisfy the
causality condition at the center and should be mono-
tonically decreasing outward besides being continuous
at the interface.
(vi) Adiabatic index: The adiabatic index should be con-
tinuous at the interface and should satisfy the Bondi
condition.
(vii) Energy conditions: The core and the envelope for the
star should satisfy the energy conditions besides being
continuous at the interface.
(viii) TOV equation: The TOV condition should be satisfied
within the star and all the three forces should be con-
tinuous at the interface resulting the system to be in
static equilibrium.
(ix) At the stellar boundary p
r
(R
E
) = 0[47].
(x) The metric potentials of the core region should match
smoothly with the gravitational potentials of the enve-
lope region [47].
(xi) The gravitational potentials of the envelope layer
should connected smoothly over the boundary with
the Schwarzschild exterior metric [47].
Fig. 1 Variation of metric potentials with radial coordinate r for (i)
SAX J1808.4-3658 (above) with mass M = 0.7M
and radius R
E
=
6.2 km for the values of b = 0.000035/km
2
, B = 0.00025/km
2
, A =
90/km
2
, α = 0.1409/km
2
, R
C
= 3.393 km; (ii) 4U1608-52 (below)
with mass M = 1.85M
and radius R
E
= 8.2 km for the values of b =
0.00003/km
2
, B = 0.00005/km
2
, A = 155/km
2
, α = 0.312/km
2
,
R
C
= 2.459 km
4 The core-envelope model
For core region (0 ≤r ≤ R
C
), we choose Tolman VII type of
g
11
for the gravitational potential e
λ
satisfying a linear EOS
z = e
−λ
C
= 1 − ax + bx
2
, (15)
p
r
C
= αρ − β, (16)
where a, b ,α, and β are constants. Substituting z value from
Eq. (15)inEq.(11) and using Eqs. 8, 9, 15 and 16 , we obtain
the following differential equation:
y
(x)
y(x)
=
P
1
+ P
2
x
2
−ax + bx
2
+ 1
. (17)
On integrating Eq. (17), we get
y = e
ν
C
= P
3
e
⎛
⎜
⎜
⎜
⎜
⎜
⎝
2
(
aP
2
+2bP
1
)
tan
−1
2bx −a
√
4b−a
2
√
4b−a
2
+P
2
log
(
−ax+bx
2
+1
)
4b
⎞
⎟
⎟
⎟
⎟
⎟
⎠
, (18)
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