The main equations used by emacss. The first section defines the differential equations that calculate the evolution of the measurable quantities (e.g. N, |$\bar{m}$|, and rh). The second section defines the dimensionless factors used in these equations, while the third section defines additional factors used by the preceding equations in each time-step. Additional parameters derived from the variables modelled by emacss are defined in the final section.
| Output properties evolved by emacss | |
| |$\dot{E} = \frac{\epsilon |E|}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the external energy of stars. Virial equilibrium is assumed throughout, such that E = U/2. |
| |$\dot{N} = -\frac{\xi N}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the total number of bound stars. |
| |$\dot{r}_{\rm h}= \frac{\mu r_{\rm h}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the half-mass radius. |
| |$\dot{\bar{m}} = \frac{\gamma \bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of bound stars. |
| |$\dot{\kappa } = \frac{\lambda \kappa }{\tau _{\rm rh}^{\prime }}$| | Rate of change of the energy form factor, related to the density profile. |
| |$\dot{\mathcal {M}}= \mathcal {M}\frac{\mathcal {M}_1 - \mathcal {M}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the concentration parameter for evolving stars. Defined so as to express the efficiency of the adiabatic expansion caused by stellar evolution [i.e. |$\dot{E}/|E| = -\mathcal {M}\dot{\bar{m}}/{\bar{m}}$|, and so |$\dot{r}_{\rm h}/r_{\rm h}= -(\mathcal {M}-2)\dot{\bar{m}}/{\bar{m}}$|]. |
| |${\dot{\bar{m}}_{\rm s}}= \frac{\gamma _{\rm s}\bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of stars due only to the in situ mass loss caused by stellar evolution. If escaping stars have no preferential mass (e.g. all masses of stars are ejected), |$\bar{m}\equiv \bar{m}_{\rm s}$|. |
| Dimensionless (differential) parameters | |
| ξ = ξe + ξi | Dimensionless escape rate per |$\tau _{\rm rh}^{\prime }$| due to both direct and induced mechanisms. |
| |$\xi _{\rm e}= \mathcal {F}\xi _0(1-\mathcal {P})+\left[f+(1-f)\mathcal {F}\right]\frac{3}{5}\zeta \mathcal {P}$| | Direct relaxation driven escape rate. The first term comes from internal effects (Baumgardt et al. 2002) and the second from mass loss due to a tidal field (Paper I and references therein). |
| ξi = findγs | Dynamical escape rate per |$\tau _{\rm rh}^{\prime }$| induced by stellar evolution. |
| μ = ϵ − 2ξ + 2γ + λ | Dimensionless change of rh per |$\tau _{\rm rh}^{\prime }$| due to dynamical and stellar evolution. Responds so as to maintain the balance of energy (i.e. by conservation of energy). |
| γ = γs + γe | Dimensionless change in |$\bar{m}$| due to both stellar evolution and the preferential ejection of low-mass stars. |
| |$\gamma _{\rm s}= -\frac{\nu \tau _{\rm rh}^{\prime }}{t}\frac{\bar{m}_{\rm s}}{\bar{m}}$| | Dimensionless change in |$\bar{m}$| due to stellar evolution. |
| |$\gamma _{\rm e}= \left[1-\frac{m_{\rm esc}}{\bar{m}}\right]\mathcal {S}\mathcal {U}\xi$| | Dimensionless change in |$\bar{m}$| due to the preferential ejection of low-mass stars. |
| $\lambda = \left\{\begin{matrix}
0, & n\,{<}\,n_{c}/2,\\
(\kappa _{1}-\kappa )\left ( \frac{2n}{n_{c}}-1 \right ), & {\rm otherwise,}
\end{matrix}\right.$ | Dimensionless change in the energy form factor, most significant just prior to core collapse (e.g. see Paper II). In Paper II an alternative definition is used to represent the gravothermal catastrophe. However, since SCs with MFs do not undergo the gravothermal catastrophe, we use an approximate form in this study. |
| Variable factors | |
| |$\mathcal {P}= \left(\frac{\mathcal {R}_{\rm hJ}}{\mathcal {R}_{1}}\right)^z\left[\frac{N\log (\gamma _{\rm c}N_1)}{N_1\log (\gamma _{\rm c}N)}\right]^{1-x}$| | Function parametrising the rate of escape due to a tidal field (Paper I). |
| $\mathcal {F}= \left\lbrace \begin{array}{@{}ll@{}}
0, & \,\,\, n {<} n_{\rm c}/2,\\
\left(\frac{2n}{n_{\rm c}}-1\right), & \,\,\, n_{\rm c}/2 \le n \le n_{\rm c},\\
1, & \,\,\, n_{\rm c}{<} n, \end{array}\right.$ | Smoothing factor to connect the escape rate in unbalanced evolution to the escape rate in balanced evolution. In Paper II it was assumed |$\mathcal {F}= \mathcal {R}_{\rm ch}^{\rm min}/\mathcal {R}_{\rm ch}$|, although rc is not available for this case. We therefore use an approximation that behaves in an equivalent manner. |
| |$\tau _{\rm e}(m) = \tau _{\rm e}(m_{\rm up})\left[1+\frac{\ln (m/m_{\rm up})}{ \ln (m_{\rm up}/m_{\rm up}^{\infty })}\right]^{a}$| | Time before the supernova of the most massive star, as a function of the upper limit of the MF (mup). Based on the analytic description of Hurley et al. (2000): see Fig. 3. This function can be inverted to give mup(t) as a function of t. |
| |$\mathcal {S}= [(\mathcal {M}-3)/(\mathcal {M}_1-3)]^q$| | Factor relating the mass segregation to the depletion of low-mass stars. |
| |$\mathcal {U}= (m_{\rm up}(t)-\bar{m})/(m_{\rm up}(t))$| | Factor to ensure |$\bar{m}{<} m_{\rm up}(t)$| at all times. |
| |$m_{\rm esc}= \mathcal {X}\left(\bar{m}-m_{\rm low}\right)+m_{\rm low}$| | (Average) mass of an escaping star. |
| ${f}_{\rm ind}= \left\lbrace \begin{array}{@{}ll@{}}
\mathcal {Y}\left(\mathcal {R}_{\rm hJ}-\mathcal {R}_{1}\right)^b, & \mathcal {R}_{\rm hJ}> \mathcal {R}_{1}, \\
0, & {\rm otherwise.} \end{array}\right.$ | Approximation for the relationship between induced escape to mass loss through stellar evolution (see Lamers et al. 2010). |
| $\psi (t) = \left\lbrace \begin{array}{@{}ll@{}}
\psi _{\rm 1}, & \,\,\, t \le \tau _{\rm e}, \\
(\psi _{\rm 1}-\psi _{\rm 0})\left[\frac{t}{\tau _{\rm e}(m_{\rm up})}\right]^{y}+\psi _{\rm 0}, & \,\,\, t {>} \tau _{\rm e}.
\end{array}\right.$ | Modification to the standard definition of τrh, adjusting to account for the presence of a MF. Approximately represents |$\psi = \overline{m^{\beta }}/\overline{m}^{\beta }$|: see (Spitzer & Hart 1971). |
| Derived cluster properties | |
| |$\tau _{\rm rh}= 0.138\frac{\left(Nr_{\rm h}^3\right)^{1/2}}{\sqrt{G\bar{m}}\log (\gamma _{\rm c}N)}$| | Mean relaxation time of stars within the half-mass radius (Spitzer 1987). |
| |$\tau _{\rm rh}^{\prime } = \frac{\tau _{\rm rh}}{\psi (t)}$| | Modified relaxation time of stars within the half-mass radius, adjusted to consider the presence of a MF. See (Spitzer & Hart 1971). |
| |$n= \int ^{t}_0 \frac{{\rm d}t}{\tau _{\rm rh}^{\prime }}$| | Number of modified relaxation times that have elapsed at time t. |
| |$r_{\rm J}= R_{\rm G}\left[\frac{N\bar{m}}{2M_{\rm G}(<R_{\rm G})}\right]^{\frac{1}{3}}$| | Jacobi (tidal) radius for an isothermal galaxy halo. For a point-mass galaxy the factor of 2 in the denominator is replaced by a factor of 3. The mass contained within the galactocentric (orbital) radius is defined by MG(<RG). |
| Output properties evolved by emacss | |
| |$\dot{E} = \frac{\epsilon |E|}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the external energy of stars. Virial equilibrium is assumed throughout, such that E = U/2. |
| |$\dot{N} = -\frac{\xi N}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the total number of bound stars. |
| |$\dot{r}_{\rm h}= \frac{\mu r_{\rm h}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the half-mass radius. |
| |$\dot{\bar{m}} = \frac{\gamma \bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of bound stars. |
| |$\dot{\kappa } = \frac{\lambda \kappa }{\tau _{\rm rh}^{\prime }}$| | Rate of change of the energy form factor, related to the density profile. |
| |$\dot{\mathcal {M}}= \mathcal {M}\frac{\mathcal {M}_1 - \mathcal {M}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the concentration parameter for evolving stars. Defined so as to express the efficiency of the adiabatic expansion caused by stellar evolution [i.e. |$\dot{E}/|E| = -\mathcal {M}\dot{\bar{m}}/{\bar{m}}$|, and so |$\dot{r}_{\rm h}/r_{\rm h}= -(\mathcal {M}-2)\dot{\bar{m}}/{\bar{m}}$|]. |
| |${\dot{\bar{m}}_{\rm s}}= \frac{\gamma _{\rm s}\bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of stars due only to the in situ mass loss caused by stellar evolution. If escaping stars have no preferential mass (e.g. all masses of stars are ejected), |$\bar{m}\equiv \bar{m}_{\rm s}$|. |
| Dimensionless (differential) parameters | |
| ξ = ξe + ξi | Dimensionless escape rate per |$\tau _{\rm rh}^{\prime }$| due to both direct and induced mechanisms. |
| |$\xi _{\rm e}= \mathcal {F}\xi _0(1-\mathcal {P})+\left[f+(1-f)\mathcal {F}\right]\frac{3}{5}\zeta \mathcal {P}$| | Direct relaxation driven escape rate. The first term comes from internal effects (Baumgardt et al. 2002) and the second from mass loss due to a tidal field (Paper I and references therein). |
| ξi = findγs | Dynamical escape rate per |$\tau _{\rm rh}^{\prime }$| induced by stellar evolution. |
| μ = ϵ − 2ξ + 2γ + λ | Dimensionless change of rh per |$\tau _{\rm rh}^{\prime }$| due to dynamical and stellar evolution. Responds so as to maintain the balance of energy (i.e. by conservation of energy). |
| γ = γs + γe | Dimensionless change in |$\bar{m}$| due to both stellar evolution and the preferential ejection of low-mass stars. |
| |$\gamma _{\rm s}= -\frac{\nu \tau _{\rm rh}^{\prime }}{t}\frac{\bar{m}_{\rm s}}{\bar{m}}$| | Dimensionless change in |$\bar{m}$| due to stellar evolution. |
| |$\gamma _{\rm e}= \left[1-\frac{m_{\rm esc}}{\bar{m}}\right]\mathcal {S}\mathcal {U}\xi$| | Dimensionless change in |$\bar{m}$| due to the preferential ejection of low-mass stars. |
| $\lambda = \left\{\begin{matrix}
0, & n\,{<}\,n_{c}/2,\\
(\kappa _{1}-\kappa )\left ( \frac{2n}{n_{c}}-1 \right ), & {\rm otherwise,}
\end{matrix}\right.$ | Dimensionless change in the energy form factor, most significant just prior to core collapse (e.g. see Paper II). In Paper II an alternative definition is used to represent the gravothermal catastrophe. However, since SCs with MFs do not undergo the gravothermal catastrophe, we use an approximate form in this study. |
| Variable factors | |
| |$\mathcal {P}= \left(\frac{\mathcal {R}_{\rm hJ}}{\mathcal {R}_{1}}\right)^z\left[\frac{N\log (\gamma _{\rm c}N_1)}{N_1\log (\gamma _{\rm c}N)}\right]^{1-x}$| | Function parametrising the rate of escape due to a tidal field (Paper I). |
| $\mathcal {F}= \left\lbrace \begin{array}{@{}ll@{}}
0, & \,\,\, n {<} n_{\rm c}/2,\\
\left(\frac{2n}{n_{\rm c}}-1\right), & \,\,\, n_{\rm c}/2 \le n \le n_{\rm c},\\
1, & \,\,\, n_{\rm c}{<} n, \end{array}\right.$ | Smoothing factor to connect the escape rate in unbalanced evolution to the escape rate in balanced evolution. In Paper II it was assumed |$\mathcal {F}= \mathcal {R}_{\rm ch}^{\rm min}/\mathcal {R}_{\rm ch}$|, although rc is not available for this case. We therefore use an approximation that behaves in an equivalent manner. |
| |$\tau _{\rm e}(m) = \tau _{\rm e}(m_{\rm up})\left[1+\frac{\ln (m/m_{\rm up})}{ \ln (m_{\rm up}/m_{\rm up}^{\infty })}\right]^{a}$| | Time before the supernova of the most massive star, as a function of the upper limit of the MF (mup). Based on the analytic description of Hurley et al. (2000): see Fig. 3. This function can be inverted to give mup(t) as a function of t. |
| |$\mathcal {S}= [(\mathcal {M}-3)/(\mathcal {M}_1-3)]^q$| | Factor relating the mass segregation to the depletion of low-mass stars. |
| |$\mathcal {U}= (m_{\rm up}(t)-\bar{m})/(m_{\rm up}(t))$| | Factor to ensure |$\bar{m}{<} m_{\rm up}(t)$| at all times. |
| |$m_{\rm esc}= \mathcal {X}\left(\bar{m}-m_{\rm low}\right)+m_{\rm low}$| | (Average) mass of an escaping star. |
| ${f}_{\rm ind}= \left\lbrace \begin{array}{@{}ll@{}}
\mathcal {Y}\left(\mathcal {R}_{\rm hJ}-\mathcal {R}_{1}\right)^b, & \mathcal {R}_{\rm hJ}> \mathcal {R}_{1}, \\
0, & {\rm otherwise.} \end{array}\right.$ | Approximation for the relationship between induced escape to mass loss through stellar evolution (see Lamers et al. 2010). |
| $\psi (t) = \left\lbrace \begin{array}{@{}ll@{}}
\psi _{\rm 1}, & \,\,\, t \le \tau _{\rm e}, \\
(\psi _{\rm 1}-\psi _{\rm 0})\left[\frac{t}{\tau _{\rm e}(m_{\rm up})}\right]^{y}+\psi _{\rm 0}, & \,\,\, t {>} \tau _{\rm e}.
\end{array}\right.$ | Modification to the standard definition of τrh, adjusting to account for the presence of a MF. Approximately represents |$\psi = \overline{m^{\beta }}/\overline{m}^{\beta }$|: see (Spitzer & Hart 1971). |
| Derived cluster properties | |
| |$\tau _{\rm rh}= 0.138\frac{\left(Nr_{\rm h}^3\right)^{1/2}}{\sqrt{G\bar{m}}\log (\gamma _{\rm c}N)}$| | Mean relaxation time of stars within the half-mass radius (Spitzer 1987). |
| |$\tau _{\rm rh}^{\prime } = \frac{\tau _{\rm rh}}{\psi (t)}$| | Modified relaxation time of stars within the half-mass radius, adjusted to consider the presence of a MF. See (Spitzer & Hart 1971). |
| |$n= \int ^{t}_0 \frac{{\rm d}t}{\tau _{\rm rh}^{\prime }}$| | Number of modified relaxation times that have elapsed at time t. |
| |$r_{\rm J}= R_{\rm G}\left[\frac{N\bar{m}}{2M_{\rm G}(<R_{\rm G})}\right]^{\frac{1}{3}}$| | Jacobi (tidal) radius for an isothermal galaxy halo. For a point-mass galaxy the factor of 2 in the denominator is replaced by a factor of 3. The mass contained within the galactocentric (orbital) radius is defined by MG(<RG). |
The main equations used by emacss. The first section defines the differential equations that calculate the evolution of the measurable quantities (e.g. N, |$\bar{m}$|, and rh). The second section defines the dimensionless factors used in these equations, while the third section defines additional factors used by the preceding equations in each time-step. Additional parameters derived from the variables modelled by emacss are defined in the final section.
| Output properties evolved by emacss | |
| |$\dot{E} = \frac{\epsilon |E|}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the external energy of stars. Virial equilibrium is assumed throughout, such that E = U/2. |
| |$\dot{N} = -\frac{\xi N}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the total number of bound stars. |
| |$\dot{r}_{\rm h}= \frac{\mu r_{\rm h}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the half-mass radius. |
| |$\dot{\bar{m}} = \frac{\gamma \bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of bound stars. |
| |$\dot{\kappa } = \frac{\lambda \kappa }{\tau _{\rm rh}^{\prime }}$| | Rate of change of the energy form factor, related to the density profile. |
| |$\dot{\mathcal {M}}= \mathcal {M}\frac{\mathcal {M}_1 - \mathcal {M}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the concentration parameter for evolving stars. Defined so as to express the efficiency of the adiabatic expansion caused by stellar evolution [i.e. |$\dot{E}/|E| = -\mathcal {M}\dot{\bar{m}}/{\bar{m}}$|, and so |$\dot{r}_{\rm h}/r_{\rm h}= -(\mathcal {M}-2)\dot{\bar{m}}/{\bar{m}}$|]. |
| |${\dot{\bar{m}}_{\rm s}}= \frac{\gamma _{\rm s}\bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of stars due only to the in situ mass loss caused by stellar evolution. If escaping stars have no preferential mass (e.g. all masses of stars are ejected), |$\bar{m}\equiv \bar{m}_{\rm s}$|. |
| Dimensionless (differential) parameters | |
| ξ = ξe + ξi | Dimensionless escape rate per |$\tau _{\rm rh}^{\prime }$| due to both direct and induced mechanisms. |
| |$\xi _{\rm e}= \mathcal {F}\xi _0(1-\mathcal {P})+\left[f+(1-f)\mathcal {F}\right]\frac{3}{5}\zeta \mathcal {P}$| | Direct relaxation driven escape rate. The first term comes from internal effects (Baumgardt et al. 2002) and the second from mass loss due to a tidal field (Paper I and references therein). |
| ξi = findγs | Dynamical escape rate per |$\tau _{\rm rh}^{\prime }$| induced by stellar evolution. |
| μ = ϵ − 2ξ + 2γ + λ | Dimensionless change of rh per |$\tau _{\rm rh}^{\prime }$| due to dynamical and stellar evolution. Responds so as to maintain the balance of energy (i.e. by conservation of energy). |
| γ = γs + γe | Dimensionless change in |$\bar{m}$| due to both stellar evolution and the preferential ejection of low-mass stars. |
| |$\gamma _{\rm s}= -\frac{\nu \tau _{\rm rh}^{\prime }}{t}\frac{\bar{m}_{\rm s}}{\bar{m}}$| | Dimensionless change in |$\bar{m}$| due to stellar evolution. |
| |$\gamma _{\rm e}= \left[1-\frac{m_{\rm esc}}{\bar{m}}\right]\mathcal {S}\mathcal {U}\xi$| | Dimensionless change in |$\bar{m}$| due to the preferential ejection of low-mass stars. |
| $\lambda = \left\{\begin{matrix}
0, & n\,{<}\,n_{c}/2,\\
(\kappa _{1}-\kappa )\left ( \frac{2n}{n_{c}}-1 \right ), & {\rm otherwise,}
\end{matrix}\right.$ | Dimensionless change in the energy form factor, most significant just prior to core collapse (e.g. see Paper II). In Paper II an alternative definition is used to represent the gravothermal catastrophe. However, since SCs with MFs do not undergo the gravothermal catastrophe, we use an approximate form in this study. |
| Variable factors | |
| |$\mathcal {P}= \left(\frac{\mathcal {R}_{\rm hJ}}{\mathcal {R}_{1}}\right)^z\left[\frac{N\log (\gamma _{\rm c}N_1)}{N_1\log (\gamma _{\rm c}N)}\right]^{1-x}$| | Function parametrising the rate of escape due to a tidal field (Paper I). |
| $\mathcal {F}= \left\lbrace \begin{array}{@{}ll@{}}
0, & \,\,\, n {<} n_{\rm c}/2,\\
\left(\frac{2n}{n_{\rm c}}-1\right), & \,\,\, n_{\rm c}/2 \le n \le n_{\rm c},\\
1, & \,\,\, n_{\rm c}{<} n, \end{array}\right.$ | Smoothing factor to connect the escape rate in unbalanced evolution to the escape rate in balanced evolution. In Paper II it was assumed |$\mathcal {F}= \mathcal {R}_{\rm ch}^{\rm min}/\mathcal {R}_{\rm ch}$|, although rc is not available for this case. We therefore use an approximation that behaves in an equivalent manner. |
| |$\tau _{\rm e}(m) = \tau _{\rm e}(m_{\rm up})\left[1+\frac{\ln (m/m_{\rm up})}{ \ln (m_{\rm up}/m_{\rm up}^{\infty })}\right]^{a}$| | Time before the supernova of the most massive star, as a function of the upper limit of the MF (mup). Based on the analytic description of Hurley et al. (2000): see Fig. 3. This function can be inverted to give mup(t) as a function of t. |
| |$\mathcal {S}= [(\mathcal {M}-3)/(\mathcal {M}_1-3)]^q$| | Factor relating the mass segregation to the depletion of low-mass stars. |
| |$\mathcal {U}= (m_{\rm up}(t)-\bar{m})/(m_{\rm up}(t))$| | Factor to ensure |$\bar{m}{<} m_{\rm up}(t)$| at all times. |
| |$m_{\rm esc}= \mathcal {X}\left(\bar{m}-m_{\rm low}\right)+m_{\rm low}$| | (Average) mass of an escaping star. |
| ${f}_{\rm ind}= \left\lbrace \begin{array}{@{}ll@{}}
\mathcal {Y}\left(\mathcal {R}_{\rm hJ}-\mathcal {R}_{1}\right)^b, & \mathcal {R}_{\rm hJ}> \mathcal {R}_{1}, \\
0, & {\rm otherwise.} \end{array}\right.$ | Approximation for the relationship between induced escape to mass loss through stellar evolution (see Lamers et al. 2010). |
| $\psi (t) = \left\lbrace \begin{array}{@{}ll@{}}
\psi _{\rm 1}, & \,\,\, t \le \tau _{\rm e}, \\
(\psi _{\rm 1}-\psi _{\rm 0})\left[\frac{t}{\tau _{\rm e}(m_{\rm up})}\right]^{y}+\psi _{\rm 0}, & \,\,\, t {>} \tau _{\rm e}.
\end{array}\right.$ | Modification to the standard definition of τrh, adjusting to account for the presence of a MF. Approximately represents |$\psi = \overline{m^{\beta }}/\overline{m}^{\beta }$|: see (Spitzer & Hart 1971). |
| Derived cluster properties | |
| |$\tau _{\rm rh}= 0.138\frac{\left(Nr_{\rm h}^3\right)^{1/2}}{\sqrt{G\bar{m}}\log (\gamma _{\rm c}N)}$| | Mean relaxation time of stars within the half-mass radius (Spitzer 1987). |
| |$\tau _{\rm rh}^{\prime } = \frac{\tau _{\rm rh}}{\psi (t)}$| | Modified relaxation time of stars within the half-mass radius, adjusted to consider the presence of a MF. See (Spitzer & Hart 1971). |
| |$n= \int ^{t}_0 \frac{{\rm d}t}{\tau _{\rm rh}^{\prime }}$| | Number of modified relaxation times that have elapsed at time t. |
| |$r_{\rm J}= R_{\rm G}\left[\frac{N\bar{m}}{2M_{\rm G}(<R_{\rm G})}\right]^{\frac{1}{3}}$| | Jacobi (tidal) radius for an isothermal galaxy halo. For a point-mass galaxy the factor of 2 in the denominator is replaced by a factor of 3. The mass contained within the galactocentric (orbital) radius is defined by MG(<RG). |
| Output properties evolved by emacss | |
| |$\dot{E} = \frac{\epsilon |E|}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the external energy of stars. Virial equilibrium is assumed throughout, such that E = U/2. |
| |$\dot{N} = -\frac{\xi N}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the total number of bound stars. |
| |$\dot{r}_{\rm h}= \frac{\mu r_{\rm h}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the half-mass radius. |
| |$\dot{\bar{m}} = \frac{\gamma \bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of bound stars. |
| |$\dot{\kappa } = \frac{\lambda \kappa }{\tau _{\rm rh}^{\prime }}$| | Rate of change of the energy form factor, related to the density profile. |
| |$\dot{\mathcal {M}}= \mathcal {M}\frac{\mathcal {M}_1 - \mathcal {M}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the concentration parameter for evolving stars. Defined so as to express the efficiency of the adiabatic expansion caused by stellar evolution [i.e. |$\dot{E}/|E| = -\mathcal {M}\dot{\bar{m}}/{\bar{m}}$|, and so |$\dot{r}_{\rm h}/r_{\rm h}= -(\mathcal {M}-2)\dot{\bar{m}}/{\bar{m}}$|]. |
| |${\dot{\bar{m}}_{\rm s}}= \frac{\gamma _{\rm s}\bar{m}}{\tau _{\rm rh}^{\prime }}$| | Rate of change of the mean mass of stars due only to the in situ mass loss caused by stellar evolution. If escaping stars have no preferential mass (e.g. all masses of stars are ejected), |$\bar{m}\equiv \bar{m}_{\rm s}$|. |
| Dimensionless (differential) parameters | |
| ξ = ξe + ξi | Dimensionless escape rate per |$\tau _{\rm rh}^{\prime }$| due to both direct and induced mechanisms. |
| |$\xi _{\rm e}= \mathcal {F}\xi _0(1-\mathcal {P})+\left[f+(1-f)\mathcal {F}\right]\frac{3}{5}\zeta \mathcal {P}$| | Direct relaxation driven escape rate. The first term comes from internal effects (Baumgardt et al. 2002) and the second from mass loss due to a tidal field (Paper I and references therein). |
| ξi = findγs | Dynamical escape rate per |$\tau _{\rm rh}^{\prime }$| induced by stellar evolution. |
| μ = ϵ − 2ξ + 2γ + λ | Dimensionless change of rh per |$\tau _{\rm rh}^{\prime }$| due to dynamical and stellar evolution. Responds so as to maintain the balance of energy (i.e. by conservation of energy). |
| γ = γs + γe | Dimensionless change in |$\bar{m}$| due to both stellar evolution and the preferential ejection of low-mass stars. |
| |$\gamma _{\rm s}= -\frac{\nu \tau _{\rm rh}^{\prime }}{t}\frac{\bar{m}_{\rm s}}{\bar{m}}$| | Dimensionless change in |$\bar{m}$| due to stellar evolution. |
| |$\gamma _{\rm e}= \left[1-\frac{m_{\rm esc}}{\bar{m}}\right]\mathcal {S}\mathcal {U}\xi$| | Dimensionless change in |$\bar{m}$| due to the preferential ejection of low-mass stars. |
| $\lambda = \left\{\begin{matrix}
0, & n\,{<}\,n_{c}/2,\\
(\kappa _{1}-\kappa )\left ( \frac{2n}{n_{c}}-1 \right ), & {\rm otherwise,}
\end{matrix}\right.$ | Dimensionless change in the energy form factor, most significant just prior to core collapse (e.g. see Paper II). In Paper II an alternative definition is used to represent the gravothermal catastrophe. However, since SCs with MFs do not undergo the gravothermal catastrophe, we use an approximate form in this study. |
| Variable factors | |
| |$\mathcal {P}= \left(\frac{\mathcal {R}_{\rm hJ}}{\mathcal {R}_{1}}\right)^z\left[\frac{N\log (\gamma _{\rm c}N_1)}{N_1\log (\gamma _{\rm c}N)}\right]^{1-x}$| | Function parametrising the rate of escape due to a tidal field (Paper I). |
| $\mathcal {F}= \left\lbrace \begin{array}{@{}ll@{}}
0, & \,\,\, n {<} n_{\rm c}/2,\\
\left(\frac{2n}{n_{\rm c}}-1\right), & \,\,\, n_{\rm c}/2 \le n \le n_{\rm c},\\
1, & \,\,\, n_{\rm c}{<} n, \end{array}\right.$ | Smoothing factor to connect the escape rate in unbalanced evolution to the escape rate in balanced evolution. In Paper II it was assumed |$\mathcal {F}= \mathcal {R}_{\rm ch}^{\rm min}/\mathcal {R}_{\rm ch}$|, although rc is not available for this case. We therefore use an approximation that behaves in an equivalent manner. |
| |$\tau _{\rm e}(m) = \tau _{\rm e}(m_{\rm up})\left[1+\frac{\ln (m/m_{\rm up})}{ \ln (m_{\rm up}/m_{\rm up}^{\infty })}\right]^{a}$| | Time before the supernova of the most massive star, as a function of the upper limit of the MF (mup). Based on the analytic description of Hurley et al. (2000): see Fig. 3. This function can be inverted to give mup(t) as a function of t. |
| |$\mathcal {S}= [(\mathcal {M}-3)/(\mathcal {M}_1-3)]^q$| | Factor relating the mass segregation to the depletion of low-mass stars. |
| |$\mathcal {U}= (m_{\rm up}(t)-\bar{m})/(m_{\rm up}(t))$| | Factor to ensure |$\bar{m}{<} m_{\rm up}(t)$| at all times. |
| |$m_{\rm esc}= \mathcal {X}\left(\bar{m}-m_{\rm low}\right)+m_{\rm low}$| | (Average) mass of an escaping star. |
| ${f}_{\rm ind}= \left\lbrace \begin{array}{@{}ll@{}}
\mathcal {Y}\left(\mathcal {R}_{\rm hJ}-\mathcal {R}_{1}\right)^b, & \mathcal {R}_{\rm hJ}> \mathcal {R}_{1}, \\
0, & {\rm otherwise.} \end{array}\right.$ | Approximation for the relationship between induced escape to mass loss through stellar evolution (see Lamers et al. 2010). |
| $\psi (t) = \left\lbrace \begin{array}{@{}ll@{}}
\psi _{\rm 1}, & \,\,\, t \le \tau _{\rm e}, \\
(\psi _{\rm 1}-\psi _{\rm 0})\left[\frac{t}{\tau _{\rm e}(m_{\rm up})}\right]^{y}+\psi _{\rm 0}, & \,\,\, t {>} \tau _{\rm e}.
\end{array}\right.$ | Modification to the standard definition of τrh, adjusting to account for the presence of a MF. Approximately represents |$\psi = \overline{m^{\beta }}/\overline{m}^{\beta }$|: see (Spitzer & Hart 1971). |
| Derived cluster properties | |
| |$\tau _{\rm rh}= 0.138\frac{\left(Nr_{\rm h}^3\right)^{1/2}}{\sqrt{G\bar{m}}\log (\gamma _{\rm c}N)}$| | Mean relaxation time of stars within the half-mass radius (Spitzer 1987). |
| |$\tau _{\rm rh}^{\prime } = \frac{\tau _{\rm rh}}{\psi (t)}$| | Modified relaxation time of stars within the half-mass radius, adjusted to consider the presence of a MF. See (Spitzer & Hart 1971). |
| |$n= \int ^{t}_0 \frac{{\rm d}t}{\tau _{\rm rh}^{\prime }}$| | Number of modified relaxation times that have elapsed at time t. |
| |$r_{\rm J}= R_{\rm G}\left[\frac{N\bar{m}}{2M_{\rm G}(<R_{\rm G})}\right]^{\frac{1}{3}}$| | Jacobi (tidal) radius for an isothermal galaxy halo. For a point-mass galaxy the factor of 2 in the denominator is replaced by a factor of 3. The mass contained within the galactocentric (orbital) radius is defined by MG(<RG). |
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