Non-radial oscillations of hadronic neutron stars, quark stars, and hybrid stars: calculation of f, p, and g mode frequencies

Abstract The composition and equation of state (EoS) of dense matter relevant to compact stars are quite inconclusive. However, certain observational constraints on the structural properties of compact stars help us constrain the EoS to a fair extent. Moreover, gravitational asteroseismology gives a notion of the composition and EoS of compact stars. The next generation gravitational wave (GW) detectors are likely to detect several oscillation mode frequencies of the GWs. In this work we compute the fundamental ( f ) and the first pressure ( $$p_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ) mode frequencies ( $$f_f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> and $$f_{p1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> , respectively) with different compositions viz., hadronic, quark, and hybrid star (HS) matter. For HSs, we also study the gravity ( g ) mode frequency ( $$f_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> ). For each phase we also study the correlation between the oscillation frequencies of 1.4 $$M_{\odot }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:math> and 2.01 $$M_{\odot }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:math> compact stars with other different properties. We find that various possible composition of compact stars substantially affects the oscillation frequencies. However, the mass-scaled angular f mode frequency ( $$\omega _f M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> ) varies universally with compactness ( C ) for all hadronic, quark and hybrid stars. The f mode frequency ( $$f_{f_{1.4}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:msub> </mml:math> ) of the canonical 1.4 $$M_{\odot }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:math> compact star, obtained with different composition, is quite correlated with the canonical radius ( $$R_{1.4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:math> ) and tidal deformability ( $$\varLambda _{1.4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:math> ) while $$f_{p_{1.4}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:msub> </mml:math> is well correlated with slope parameter of the symmetry energy. We also show that $$f_{g_{1.4}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:msub> </mml:math> of the HSs varies almost linearly with $$\varLambda _{1.4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:msub> </mml:math> . Should g modes be detected, they could not only support the existence of HSs, but $$f_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> could be useful to understand the strength of quark repulsion in HSs.