Overview of equations for Newtonian gravity, a Plummer softened gravitational potential, and a Wendland C2 softened potential.
| Newtonian (no softening) . | |||
|---|---|---|---|
| Potential | φN(r) | = | −GMr−1 |
| Acceleration | |aN(r)| | = | GMr−2 |
| Free-fall time | tff, N | = | |$\left(\frac{3\pi }{32 G \rho }\right)^{1/2} = 4.4\, \mathrm{Myr} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans length | λJ, N | = | |$\left(\frac{3\pi \gamma X_{\mathrm{H}}k_{\mathrm{B}}T}{32 G m_{\mathrm{H}}^2 n_{\mathrm{H}}}\right)^{1/2} = 1.5\, \mathrm{pc} \left(\frac{T}{10\, \mathrm{K}}\right)^{1/2} \left(\frac{n_{\mathrm{H}}}{100 \, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans mass | MJ, N | = | |$\frac{4\pi \rho }{3} \lambda _{\mathrm{J,N}}^3 = 46 \, \mathrm{M}_{\odot } \, \left(\frac{T}{10\, \mathrm{K}} \right)^{3/2} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Plummer softening with softening scale ϵ = ϵPlummer | |||
| Potential | φP(r, ϵ) | = | −GM(r2 + ϵ2)−1/2 |
| Acceleration | |aP(r, ϵ)| | = | GMr(r2 + ϵ2)−3/2 |
| Free-fall time | tff, P, fit | = | |$t_{\mathrm{ff,N}} \left(1 + 2^{2/3} \left(\frac{\epsilon }{R} \right)^{2} \right)^{3/4}$| |
| Jeans length | λJ, P, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2} \right)^{2/5}$| |
| Jeans mass | MJ, P, fit | = | |$M_{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2}\right)^{6/5}$| |
| Wendland C2/Swift softening with softening scale H = 3ϵ = ϵPlummer and u = r/H | |||
| Potential | φW(r < H, H) | = | −GMH−1W(u) |
| with W(u) | = | (− 3u7 + 15u6 − 28u5 + 21u4 − 7u2 + 3) | |
| φW(r ≥ H) | = | −GMr−1 | |
| Acceleration | |aW(r < H, H)| | = | GMrH−3V(u) |
| with V(u) | = | −W′(u)/u = (21u5 − 90u4 + 140u3 − 84u2 + 14) | |
| |aW(r ≥ H)| | = | GMr−2 | |
| Free-fall time | tff, W, fit | = | |$t_{\mathrm{ff,N}} \left(1 + \frac{1}{7} \left(\frac{H}{R} \right)^{3} \right)^{1/2}$| = |$t_{\mathrm{ff,N}} \left(1 + \frac{27}{7} \left(\frac{\epsilon }{R} \right)^{3} \right)^{1/2}$| |
| Jeans length | λJ, W, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| = |$\lambda _{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| |
| Jeans mass | MJ, W, fit | = | |$M_{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| = |$M_{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| |
| Newtonian (no softening) . | |||
|---|---|---|---|
| Potential | φN(r) | = | −GMr−1 |
| Acceleration | |aN(r)| | = | GMr−2 |
| Free-fall time | tff, N | = | |$\left(\frac{3\pi }{32 G \rho }\right)^{1/2} = 4.4\, \mathrm{Myr} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans length | λJ, N | = | |$\left(\frac{3\pi \gamma X_{\mathrm{H}}k_{\mathrm{B}}T}{32 G m_{\mathrm{H}}^2 n_{\mathrm{H}}}\right)^{1/2} = 1.5\, \mathrm{pc} \left(\frac{T}{10\, \mathrm{K}}\right)^{1/2} \left(\frac{n_{\mathrm{H}}}{100 \, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans mass | MJ, N | = | |$\frac{4\pi \rho }{3} \lambda _{\mathrm{J,N}}^3 = 46 \, \mathrm{M}_{\odot } \, \left(\frac{T}{10\, \mathrm{K}} \right)^{3/2} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Plummer softening with softening scale ϵ = ϵPlummer | |||
| Potential | φP(r, ϵ) | = | −GM(r2 + ϵ2)−1/2 |
| Acceleration | |aP(r, ϵ)| | = | GMr(r2 + ϵ2)−3/2 |
| Free-fall time | tff, P, fit | = | |$t_{\mathrm{ff,N}} \left(1 + 2^{2/3} \left(\frac{\epsilon }{R} \right)^{2} \right)^{3/4}$| |
| Jeans length | λJ, P, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2} \right)^{2/5}$| |
| Jeans mass | MJ, P, fit | = | |$M_{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2}\right)^{6/5}$| |
| Wendland C2/Swift softening with softening scale H = 3ϵ = ϵPlummer and u = r/H | |||
| Potential | φW(r < H, H) | = | −GMH−1W(u) |
| with W(u) | = | (− 3u7 + 15u6 − 28u5 + 21u4 − 7u2 + 3) | |
| φW(r ≥ H) | = | −GMr−1 | |
| Acceleration | |aW(r < H, H)| | = | GMrH−3V(u) |
| with V(u) | = | −W′(u)/u = (21u5 − 90u4 + 140u3 − 84u2 + 14) | |
| |aW(r ≥ H)| | = | GMr−2 | |
| Free-fall time | tff, W, fit | = | |$t_{\mathrm{ff,N}} \left(1 + \frac{1}{7} \left(\frac{H}{R} \right)^{3} \right)^{1/2}$| = |$t_{\mathrm{ff,N}} \left(1 + \frac{27}{7} \left(\frac{\epsilon }{R} \right)^{3} \right)^{1/2}$| |
| Jeans length | λJ, W, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| = |$\lambda _{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| |
| Jeans mass | MJ, W, fit | = | |$M_{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| = |$M_{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| |
Overview of equations for Newtonian gravity, a Plummer softened gravitational potential, and a Wendland C2 softened potential.
| Newtonian (no softening) . | |||
|---|---|---|---|
| Potential | φN(r) | = | −GMr−1 |
| Acceleration | |aN(r)| | = | GMr−2 |
| Free-fall time | tff, N | = | |$\left(\frac{3\pi }{32 G \rho }\right)^{1/2} = 4.4\, \mathrm{Myr} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans length | λJ, N | = | |$\left(\frac{3\pi \gamma X_{\mathrm{H}}k_{\mathrm{B}}T}{32 G m_{\mathrm{H}}^2 n_{\mathrm{H}}}\right)^{1/2} = 1.5\, \mathrm{pc} \left(\frac{T}{10\, \mathrm{K}}\right)^{1/2} \left(\frac{n_{\mathrm{H}}}{100 \, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans mass | MJ, N | = | |$\frac{4\pi \rho }{3} \lambda _{\mathrm{J,N}}^3 = 46 \, \mathrm{M}_{\odot } \, \left(\frac{T}{10\, \mathrm{K}} \right)^{3/2} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Plummer softening with softening scale ϵ = ϵPlummer | |||
| Potential | φP(r, ϵ) | = | −GM(r2 + ϵ2)−1/2 |
| Acceleration | |aP(r, ϵ)| | = | GMr(r2 + ϵ2)−3/2 |
| Free-fall time | tff, P, fit | = | |$t_{\mathrm{ff,N}} \left(1 + 2^{2/3} \left(\frac{\epsilon }{R} \right)^{2} \right)^{3/4}$| |
| Jeans length | λJ, P, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2} \right)^{2/5}$| |
| Jeans mass | MJ, P, fit | = | |$M_{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2}\right)^{6/5}$| |
| Wendland C2/Swift softening with softening scale H = 3ϵ = ϵPlummer and u = r/H | |||
| Potential | φW(r < H, H) | = | −GMH−1W(u) |
| with W(u) | = | (− 3u7 + 15u6 − 28u5 + 21u4 − 7u2 + 3) | |
| φW(r ≥ H) | = | −GMr−1 | |
| Acceleration | |aW(r < H, H)| | = | GMrH−3V(u) |
| with V(u) | = | −W′(u)/u = (21u5 − 90u4 + 140u3 − 84u2 + 14) | |
| |aW(r ≥ H)| | = | GMr−2 | |
| Free-fall time | tff, W, fit | = | |$t_{\mathrm{ff,N}} \left(1 + \frac{1}{7} \left(\frac{H}{R} \right)^{3} \right)^{1/2}$| = |$t_{\mathrm{ff,N}} \left(1 + \frac{27}{7} \left(\frac{\epsilon }{R} \right)^{3} \right)^{1/2}$| |
| Jeans length | λJ, W, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| = |$\lambda _{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| |
| Jeans mass | MJ, W, fit | = | |$M_{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| = |$M_{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| |
| Newtonian (no softening) . | |||
|---|---|---|---|
| Potential | φN(r) | = | −GMr−1 |
| Acceleration | |aN(r)| | = | GMr−2 |
| Free-fall time | tff, N | = | |$\left(\frac{3\pi }{32 G \rho }\right)^{1/2} = 4.4\, \mathrm{Myr} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans length | λJ, N | = | |$\left(\frac{3\pi \gamma X_{\mathrm{H}}k_{\mathrm{B}}T}{32 G m_{\mathrm{H}}^2 n_{\mathrm{H}}}\right)^{1/2} = 1.5\, \mathrm{pc} \left(\frac{T}{10\, \mathrm{K}}\right)^{1/2} \left(\frac{n_{\mathrm{H}}}{100 \, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Jeans mass | MJ, N | = | |$\frac{4\pi \rho }{3} \lambda _{\mathrm{J,N}}^3 = 46 \, \mathrm{M}_{\odot } \, \left(\frac{T}{10\, \mathrm{K}} \right)^{3/2} \left(\frac{n_{\mathrm{H}}}{100\, \mathrm{cm}^{-3}}\right)^{-1/2}$| |
| Plummer softening with softening scale ϵ = ϵPlummer | |||
| Potential | φP(r, ϵ) | = | −GM(r2 + ϵ2)−1/2 |
| Acceleration | |aP(r, ϵ)| | = | GMr(r2 + ϵ2)−3/2 |
| Free-fall time | tff, P, fit | = | |$t_{\mathrm{ff,N}} \left(1 + 2^{2/3} \left(\frac{\epsilon }{R} \right)^{2} \right)^{3/4}$| |
| Jeans length | λJ, P, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2} \right)^{2/5}$| |
| Jeans mass | MJ, P, fit | = | |$M_{\mathrm{J,N}} \left(1 + 1.42 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^{3/2}\right)^{6/5}$| |
| Wendland C2/Swift softening with softening scale H = 3ϵ = ϵPlummer and u = r/H | |||
| Potential | φW(r < H, H) | = | −GMH−1W(u) |
| with W(u) | = | (− 3u7 + 15u6 − 28u5 + 21u4 − 7u2 + 3) | |
| φW(r ≥ H) | = | −GMr−1 | |
| Acceleration | |aW(r < H, H)| | = | GMrH−3V(u) |
| with V(u) | = | −W′(u)/u = (21u5 − 90u4 + 140u3 − 84u2 + 14) | |
| |aW(r ≥ H)| | = | GMr−2 | |
| Free-fall time | tff, W, fit | = | |$t_{\mathrm{ff,N}} \left(1 + \frac{1}{7} \left(\frac{H}{R} \right)^{3} \right)^{1/2}$| = |$t_{\mathrm{ff,N}} \left(1 + \frac{27}{7} \left(\frac{\epsilon }{R} \right)^{3} \right)^{1/2}$| |
| Jeans length | λJ, W, fit | = | |$\lambda _{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| = |$\lambda _{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2 \right)^{3/10}$| |
| Jeans mass | MJ, W, fit | = | |$M_{\mathrm{J,N}} \left(1 + 0.27 \left(\frac{H}{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| = |$M_{\mathrm{J,N}} \left(1 + 2.43 \left(\frac{\epsilon }{\lambda _{\mathrm{J,N}}}\right)^2\right)^{9/10}$| |
This PDF is available to Subscribers Only
View Article Abstract & Purchase OptionsFor full access to this pdf, sign in to an existing account, or purchase an annual subscription.
