Appendices

A. List of symbols

Symbol	Description
\(t\)	Time coordinate in Boyer-Linquist type coordinates
\(r\)	Radial coordinate in Boyer-Linquist type coordinates
\(\theta\)	Polar angle in Boyer-Linquist type coordinates
\(\phi\)	Azimuthal angle in Boyer-Linquist type coordinates
\(\mathsf{h}\)	Enthalpy per rest mass-energy
\(h\)	Pseudo-enthalpy defined as \(h = \ln \mathsf{h}\). Independent integration variable
\(h_{c}\)	Pseudo-enthalpy defined exactly at the center of the NS
\(h_{\epsilon}\)	Pseudo-enthalpy at the small core of radius \(\epsilon\) from the center of the NS
\(R\)	Radial coordinate in Hartle-Thorne coordinates
\(\Theta\)	Polar angle in Hartle-Thorne coordinates (\(\Theta \equiv \theta\))
\(R_{\epsilon}\)	Radial coordinate in Hartle-Thorne coordinates at small core of radius \(\epsilon\)
\(\epsilon\)	Book-keeping parameter of the Hartle-Thorne approximation
\(g_{\mu \nu}\)	Axially symmetric full spacetime metric tensor
\(g^{(0)}_{\mu \nu}\)	Spherically symmetric background metric tensor
\(h_{\mu \nu}\)	Metric perturbation tensor
\(u^{\mu}\)	Fluid 4-velocity vector of the NS
\(T_{\mu \nu}\)	Perfect fluid stress-energy tensor
\(\nu\)	Background metric function found in \(g_{tt}\)
\(\nu^{\textsf{ext}}\)	Function \(\nu\) valid only outside the NS
\(\lambda\)	Background metric function found in \(g_{rr}\)
\(\lambda^{\textsf{ext}}\)	Function \(\lambda\) defined only outside the NS
\(M\)	Enclosed mass-energy function related to \(\lambda\) as \(M \equiv (1-e^{-\lambda})R/2\)
\(f\)	Schwarzschild function given by \(f \equiv 1-2M/R\)
\(p\)	Pressure function at the interior of the NS
\(p_{c}\)	Pressure at exactly the center of the NS
\(\varepsilon\)	Total energy density function at the interior of the NS
\(M_{\star}\)	Total mass-energy of the NS
\(R_{\star}\)	Radius of the NS
\(C\)	Compactness of the NS defined as \(C \equiv M_{\star} / R_{\star}\)
\(\Omega\)	Constant angular speed of the NS
\(\omega^{(1)}\)	Metric function perturbation at \(\mathcal{O}(\epsilon)\) found in \(g_{t\phi}\)
\(\varpi^{(1)}\)	Relative angular velocity at order \(\mathcal{O}(\epsilon)\) defined as \(\varpi^{(1)} \equiv \Omega - \omega^{(1)}\)
\(\varpi^{(1)}_{1}\)	Mode \(\ell = 1\) function from the vector harmonic decomposition of \(\varpi^{(1)}\)
\(\varpi^{(1)}_{1,c}\)	Defined as \(\varpi^{(1)}_{1}\) evaluated exactly at the center of the NS
\(\varpi^{(1)}_{1,ext}\)	Function \(\varpi^{(1)}_{1}\) valid only at the exterior of the NS
\(S\)	Angular momentum of the NS
\(I\)	Moment of inertia of the NS
\(\bar{I}\)	Dimensionless moment of inertia defined as \(\bar{I} \equiv I / M^{3}_{\star}\)
\(\xi^{(2)}\)	Radial displacement function away from sphericity
\(\xi^{(2)}_{0}\)	Mode \(\ell = 0\) function from harmonic decomposition of \(\xi^{(2)}\) at \(\mathcal{O}(\epsilon^{2})\)
\(\xi^{(2)}_{2}\)	Mode \(\ell = 2\) function from harmonic decomposition of \(\xi^{(2)}\) at \(\mathcal{O}(\epsilon^{2})\)
\(\xi^{(2)}_{0,c}\)	Defined as \(\xi^{(2)}_{0}\) evaluated exactly at the center of the NS
\(h^{(2)}\)	Metric function perturbation found in \(g_{tt}\) at \(\mathcal{O}(\epsilon^{2})\)
\(h^{(2)}_{0}\)	Mode \(\ell = 0\) function from spherical harmonic decomposition of \(h^{(2)}\) at \(\mathcal{O}(\epsilon^{2})\)
\(h^{(2)}_{0,c}\)	Defined as \(h^{(2)}_{0}\) evaluated at exactly the center of the NS
\(h^{(2)}_{2}\)	Mode \(\ell = 2\) function from spherical harmonic decomposition of \(h_{2}\) at \(\mathcal{O}(\epsilon^{2})\)
\(h^{(2)}_{2,ext}\)	Function \(h^{(2)}_{2}\) defined only at the exterior of the NS
\(k^{(2)}\)	Areal radius metric function perturbation at \(\mathcal{O}(\epsilon^{2})\)
\(k^{(2)}_{2}\)	Mode \(\ell = 2\) function from harmonic decomposition of \(k^{(2)}\) at \(\mathcal{O}(\epsilon^{2})\)
\(k^{(2)}_{2,ext}\)	Function \(k^{(2)}_{2}\) defined only at the exterior of the NS
\(A\)	Integration constant which appears in \(h^{(2)}_{2,ext}\) and \(k^{(2)}_{2,ext}\)
\(B\)	Initial condition constant in \(h^{(2)}_{2}\) and \(k^{(2)}_{2}\) system of differential equations
\(m^{(2)}\)	Metric function perturbation found in \(g_{rr}\) at \(\mathcal{O}(\epsilon^{2})\)
\(m^{(2)}_{0}\)	Mode \(\ell = 0\) function from harmonic decomposition of \(m_{2}\) at \(\mathcal{O}(\epsilon^{2})\)
\(m^{(2)}_{0,ext}\)	Function \(m^{(2)}_{0}\) defined only at the exterior of the NS
\(m^{(2)}_{2}\)	Mode \(\ell = 2\) function from spherical harmonic decomposition of \(m^{(2)}\) at \(\mathcal{O}(\epsilon^{2})\)
\(m^{(2)}_{2,ext}\)	Function \(m^{(2)}_{2}\) defined only at the exterior of the NS
\(y\)	Tidal deformability function related to \(h^{(2)}_{2}\) as \(y \equiv R_{\star} h'^{(2)}_{2}/h^{(2)}_{2}\)
\(Y\)	Tidal deformability function \(y\) evaluated at \(R_{\star}\)
\(k_{2}\)	Tidal apsidal constant
\(\lambda^{\textsf{(tid)}}\)	Tidal Love number of the NS
\(\bar{\lambda}\)	Dimensionless tidal Love number
\(Q\)	Rotational quadrupole moment
\(\bar{Q}\)	Dimensionless rotational quadrupole moment
\(\delta R_{\star}\)	Monopole correction to the radius of the NS
\(\delta M_{\star}\)	Monopole correction to the mass-energy of the NS

B. Units

Geometrized units are often used in general relativity where \(G=c=1\). For neutron stars, it is convenient also to set \(M_{\odot}=1\). So, the overall units are \(G=c=M_{\odot}=1\). This is the system of units used in QLIMR. That is equivalent to make all quantities dimensionless with respect to one solar mass in units of length (\(\ell_{\odot} \sim 1.5 \, \textrm{km}\)). The following table shows the non-geometrized dimensions (NGD), geometrized dimensions (GD), neutron star dimensions (NSD) and the conversion factors between them for all the relevant quantities (Q) used in the source code.

Q	NGD	GD	NGD \(\rightarrow\) GD	GD \(\rightarrow\) NGD	NGD \(\rightarrow\) NSD	NSD \(\rightarrow\) NGD
\(\varepsilon_{c}\)	\(\mathsf{L}^{-1}\mathsf{T}^{-2}\mathsf{M}\)	\(\mathsf{L}^{-2}\)	\(G/c^{4}\)	\(c^{4}/G\)	\(G \ell^{2}_{\odot}/c^{4}\)	\(c^{4}/G \ell^{2}_{\odot}\)
\(R_{\star}\)	\(\mathsf{L}\)	\(\mathsf{L}\)	1	1	1/\(\ell_{\odot}\)	\(\ell_{\odot}\)
\(M_{\star}\)	\(\mathsf{M}\)	\(\mathsf{L}\)	\(G/c^{2}\)	\(c^{2}/G\)	\(G/\ell_{\odot}c^{2}\)	\(\ell_{\odot}c^{2}/G\)
\(I\)	\(\mathsf{L}^{2}\mathsf{M}\)	\(\mathsf{L}^{3}\)	\(G/c^{2}\)	\(c^{2}/G\)	\(G/\ell_{\odot}^{3}c^{2}\)	\(\ell_{\odot}^{3}c^{2}/G\)
\(\lambda^{\textsf{(tid)}}\)	\(\mathsf{L}^{2}\mathsf{T}^{2}\mathsf{M}\)	\(\mathsf{L}^{5}\)	\(G\)	\(1/G\)	\(G/\ell^{5}_{\odot}\)	\(\ell^{5}_{\odot}/G\)
\(Q\)	\(\mathsf{L}^{2}\mathsf{M}\)	\(\mathsf{L}^{3}\)	\(G/c^{2}\)	\(c^{2}/G\)	\(G/\ell^{3}_{\odot}c^{2}\)	\(\ell^{3}_{\odot}c^{2}/G\)
\(\delta R_{\star}\)	\(\mathsf{L}\)	\(\mathsf{L}\)	1	1	1/\(\ell_{\odot}\)	\(\ell_{\odot}\)
\(\delta M_{\star}\)	\(\mathsf{M}\)	\(\mathsf{L}\)	\(G/c^{2}\)	\(c^{2}/G\)	\(G/\ell_{\odot}c^{2}\)	\(\ell_{\odot}c^{2}/G\)
\(h\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)
\(R\)	\(\mathsf{L}\)	\(\mathsf{L}\)	1	1	1/\(\ell_{\odot}\)	\(\ell_{\odot}\)
\(\varepsilon\)	\(\mathsf{L}^{-1}\mathsf{T}^{-2}\mathsf{M}\)	\(\mathsf{L}^{-2}\)	\(G/c^{4}\)	\(c^{4}/G\)	\(G \ell_{\odot}^{2}/c^{4}\)	\(c^{4}/G \ell_{\odot}^{2}\)
\(p\)	\(\mathsf{L}^{-1}\mathsf{T}^{-2}\mathsf{M}\)	\(\mathsf{L}^{-2}\)	\(G/c^{4}\)	\(c^{4}/G\)	\(G \ell_{\odot}^{2}/c^{4}\)	\(c^{4}/G \ell_{\odot}^{2}\)
\(\nu\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)
\(Y\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)
\(\Omega\)	\(\mathsf{T}^{-1}\)	\(\mathsf{L}^{-1}\)	\(1/c\)	\(c\)	\(\ell_{\odot}/c\)	\(c/\ell_{\odot}\)
\(\varpi_{1}\)	\(\mathsf{T}^{-1}\)	\(\mathsf{L}^{-1}\)	\(1/c\)	\(c\)	\(\ell_{\odot}/c\)	\(c/\ell_{\odot}\)
\(\xi_{2}\)	\(\mathsf{L}\)	\(\mathsf{L}\)	1	1	1/\(\ell_{\odot}\)	\(\ell_{\odot}\)
\(h_{2}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)
\(K_{2}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)	\(\mathsf{-}\)
\(m_{2}\)	\(\mathsf{M}\)	\(\mathsf{L}\)	\(G/c^{2}\)	\(c^{2}/G\)	\(G/\ell_{\odot}c^{2}\)	\(\ell_{\odot}c^{2}/G\)