The effect of differential rotation on the
maximum mass of Strange Quark Stars
Magdalena Szkudlarek1
D. Gondek-Rosińska1, , L. Villain2 , M. Ansorg3
2
1
Kepler Institute of Astronomy, University of Zielona Góra, Poland
Laboratoire de Mathematiques et Physique Theorique (LMPT), Univ. F. Rebelais Tours, France
3
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Univerwitat, Jena, Germany
Warsaw, March 27th, 2017
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
1 / 11
Equation Of State (EOS)
The EOS of Strange Quark Stars
(Gondek-Rosińska et al., 2000;
Zdunik, 2002):
P = ac 2 (ρ − ρ0 )
Weber, J. Phys. 27, 465 (2001)
Magdalena Szkudlarek (IA UZ)
In general, equation of that type
corresponds to a self-bound matter at the density ρ0 at zero pressure and
√ with a fixed sound velocity ( ac 2 ).
We are considering the simplest
MIT Bag model (ms = 0), in
which above formula is exact. In
the model a = 1/3 and ρ0 =
4.2785 × 1014 gcm−3 .
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
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Rotating stars - rigid rotation
For given EoS the maximum mass of non-rotating star is uniquely determined by the TolmanOppenheimer-Volkoff equations.
Rotation can stabilize stars with masses larger than maximum mass of non-rotating star
(Baumgarte, Shapiro, Shibata, 2000, ApJ 528, L29).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
3 / 11
Rotating stars - rigid rotation
For given EoS the maximum mass of non-rotating star is uniquely determined by the TolmanOppenheimer-Volkoff equations.
Rotation can stabilize stars with masses larger than maximum mass of non-rotating star
(Baumgarte, Shapiro, Shibata, 2000, ApJ 528, L29).
It leads to the occurrence of configurations, which in the static case do not exist.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
3 / 11
Rotating stars - rigid rotation
For given EoS the maximum mass of non-rotating star is uniquely determined by the TolmanOppenheimer-Volkoff equations.
Rotation can stabilize stars with masses larger than maximum mass of non-rotating star
(Baumgarte, Shapiro, Shibata, 2000, ApJ 528, L29).
It leads to the occurrence of configurations, which in the static case do not exist.
Neutron Stars 17 % - 20 % higher masses than Mmax,stat (Cook, Shapiro, Teukolsky, 1994,
ApJ 424, 823)
Strange Quark Stars 44 % higher masses than Mmax,stat (Gondek-Rosińska et al., 2000, A&A
363, 1005; Gourgoulhon et al., 1999, A&A 349, 851).
Figure: Mass-radius relation for
static and uniformly rotating
strange stars (left) and neutron stars (right). (Gourgoulhon, Gondek-Rosińska, Haensel,
2003, A&A 412, 777)
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
3 / 11
Differential rotation
For axisymmetric simulations of strange stars we use FlatStar code (Ansorg, Gondek-Rosińska,
Villain, 2009, MNRAS 396, 4, 2359). To describe the differential rotation we use an
astrophysically motivated rotation law first presented by Komatsu et al. in 1989:
- Ω - angular velocity,
- Ωc - central angular velocity,
- A - parameter describing degree of difF (Ω) = A2 (Ωc − Ω)
ferential rotation (in our work we are using inverse parameter: Ã = re /A).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
4 / 11
Differential rotation
For axisymmetric simulations of strange stars we use FlatStar code (Ansorg, Gondek-Rosińska,
Villain, 2009, MNRAS 396, 4, 2359). To describe the differential rotation we use an
astrophysically motivated rotation law first presented by Komatsu et al. in 1989:
- Ω - angular velocity,
- Ωc - central angular velocity,
- A - parameter describing degree of difF (Ω) = A2 (Ωc − Ω)
ferential rotation (in our work we are using inverse parameter: Ã = re /A).
In this rotation law the highest angular
velocity is in the center of the star, and
it is decreasing with increasing radius.
That is why we can spin-up the stars
more than in the case of rigid rotation.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
4 / 11
Differential rotation
For axisymmetric simulations of strange stars we use FlatStar code (Ansorg, Gondek-Rosińska,
Villain, 2009, MNRAS 396, 4, 2359). To describe the differential rotation we use an
astrophysically motivated rotation law first presented by Komatsu et al. in 1989:
- Ω - angular velocity,
- Ωc - central angular velocity,
- A - parameter describing degree of difF (Ω) = A2 (Ωc − Ω)
ferential rotation (in our work we are using inverse parameter: Ã = re /A).
In this rotation law the highest angular
velocity is in the center of the star, and
it is decreasing with increasing radius.
That is why we can spin-up the stars
more than in the case of rigid rotation.
First we start from spherically symmetric configuration, then we spin-up the star with set maximum
density and degree of differential rotation - zero for uniform rotation, higher values for larger Ωc /Ωe
ratio. According to these two parameters we obtain four types of star configurations.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
4 / 11
RESULTS
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
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New types of solutions
Figure: Example of structure of the solution space showing various types of sequences with fixed à in a plane of
fixed maximal density ρmax = 0.82·1015 g /cm3 (maximal
enthalpy Hmax = 0.2). Normalized shedding parameter
β̃ in function of rp /re . Bold line corresponds to critical
value of degree of differential rotation à dividing the solution space into four regions described with type A, B, C
and D (Ansorg, Gondek-Rosińska, Villain, 2009).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
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Figure: Critical values of the degree of differential rotation à as functions of the maximum enthalpy Hmax
(bottom axis) and of the maximum density ρmax (top
axis).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
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Maximum mass of differentially rotating SQS
Figure: Highest baryonic mass MB as a function of the
maximum enthalpy Hmax (bottom axis) and of the maximum density ρmax (top axis) for fixed values of the degree of differential rotation Ã.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
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Comparison to Neutron Stars
Figure: Maximum baryonic mass normalized by the maximum mass in the non-rotating case as a function of the
degree of differential rotation Ã. Lines with dots correspond to SQS results, while other lines with other symbols
are for differentially rotating neutron stars described by
different realistic equations of states taken from Morrison
et al. (2004).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of Strange
June 16th,
Quark
2015
Stars
9 / 11
Conclusions
We performed fully relativistic simulations of axially symmetric Strange Quark
Stars for a wide range of maximum densities and degrees of differential rotation.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of June
Strange
16th,
Quark
2015Stars 10 / 11
Conclusions
We performed fully relativistic simulations of axially symmetric Strange Quark
Stars for a wide range of maximum densities and degrees of differential rotation.
We found four types of solutions: A, B, C and D. The highest increase of the
maximum mass is for type B and for low values of degree of differential rotation
(Ã).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of June
Strange
16th,
Quark
2015Stars 10 / 11
Conclusions
We performed fully relativistic simulations of axially symmetric Strange Quark
Stars for a wide range of maximum densities and degrees of differential rotation.
We found four types of solutions: A, B, C and D. The highest increase of the
maximum mass is for type B and for low values of degree of differential rotation
(Ã).
Maximum mass up to 3.6 times larger comparing to non-rotating case (about
10 M ).
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of June
Strange
16th,
Quark
2015Stars 10 / 11
Conclusions
We performed fully relativistic simulations of axially symmetric Strange Quark
Stars for a wide range of maximum densities and degrees of differential rotation.
We found four types of solutions: A, B, C and D. The highest increase of the
maximum mass is for type B and for low values of degree of differential rotation
(Ã).
Maximum mass up to 3.6 times larger comparing to non-rotating case (about
10 M ).
Strange Quark Stars has much larger increase of mass for given degree of
differential rotation (Ã) comparing to realistic EoS of Neutron Stars.
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of June
Strange
16th,
Quark
2015Stars 10 / 11
Thank You For Your Attention
Magdalena Szkudlarek (IA UZ)
The effect of differential rotation on the maximum mass of June
Strange
16th,
Quark
2015Stars 11 / 11