Numerical computation of gravitational field for general axisymmetric objects Free

Integrand of reverse Hankel transform for a bi-exponential object. Depicted is a heavy oscillation of I(k; R, z) ≡ J₀(Rk)[c exp (−|z|/c)k − exp (−k|z|)]/[(a²k² + 1)^3/2(c²k² − 1)], the integrand of the reverse Hankel transform for a bi-exponential object the volume mass density of which is expressed as ρ_BE(R, z) ≡ ρ₀ exp (−R/a − |z|/c). Plotted is the curve when a = 1, c = 0.3, R = 10, and z = 0. The wavelength of the oscillation is in proportion to 1/R, and therefore, the numerical integration of the integrand becomes impractical for large value of R, say when R > 1.

Density contour map of X/peanut-shaped object. Shown is the contour map of the volume mass density of an infinitely extended hypothetical axisymmetric object. Its volume mass density is expressed as ρ_XP(R, z) ≡ ρ₀ exp (−s/h), where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠, |$h \equiv 1 + h_6 \exp \left[ - \left( s - s_0 \right)^2/s_1^2 \right] C_6 \left( R/a, |z|/c \right)$|⁠, and C₆(ξ, η) ≡ cos [6tan ⁻¹(η/ξ)]. The figure shows a specific case when a = 1, c = 0.3, s₀ = 4, s₁ = 2, and h₆ = 0.1. The contours are drawn for every 5 per cent of the peak value at the coordinate centre. Although the object seems to be limited in a finite region, it infinitely extends with an almost exponential damping. If the unit distance is set as 10 arcsec, the contour map mimics that of the isophote of the Hubble Space Telescope/NIC3 image of NGC 128 (Ciambur & Graham 2016, fig. 5). The functional form and the parameters were experimentally determined.

Density profile of power-law disc with hole. Same as Fig. 2 but for an axisymmetric object with a central hole. The object is of a finite radial range as R_in ≤ R ≤ R_out. Its volume mass density is expressed as ρ_PH(R, z) ≡ ρ₀(R/a)^{−d − β} exp [−(z/c)²(R/a)^−2β/2], where d is a step function defined as d = d₁ when R_in < R < R₀ and d = d₂ when R₀ < R < R_out. The figure shows a specific case when a = 0.35, c = 0.105, R_in = 0.2, R₀ = 2.355, R_out = 10, d₁ = 0.53, d₂ = 8.0, and β = 1.25. This time, plotted are the contours when the relative magnitude is exponentially different as e⁻ⁿ for n = 1, 2, …, 10. Despite its outlook limited in a finite region as 0.2 < R < 2.36 and |z|/R < 0.3, the object actually extends far outwards as R < 10 and infinitely in the vertical direction when 0.2 < R < 10. If the unit distance is chosen as 20 au, this density profile is analogous to that of the observed protoplanetary disc of TW Hya (Andrews et al. 2016, fig. 2). The functional form is quoted from Hogerheijde et al. (2016) while the parameters were borrowed from a single-component determination of Andrews et al. (2016).

Density contour map of tear-drop-shaped toroid. Shown is the contour map of the volume mass density of yet another hypothetical axisymmetric object, the density of which is expressed as ρ_TD(R, z) ≡ ρ₀ exp [−p²(R/a − t)² − q²(z/c)²], where p, q, and t are auxiliary functions defined as p ≡ [1 + p₁(R/a) + p₂(R/a)²]/(1 + b₁t + b₂t²), q ≡ 1 + q₁(R/a) + q₂(R/a)², and t ≡ t₀ + t₁(z/c) + t₂(z/c)² + t₃(z/c)³. The figure shows a specific case when the parameters are set as a = 0.15, c = 0.25, b₁ = p₁/c = p₂/c² = q₁/a = 0.2, q₂/a² = 0.5, b₂ = −2, t₀ = 1, t₁ = 0, t₂/c² = −0.3, and t₃/c³ = −0.1. The contours are drawn for every 5 per cent of the peak value. The functional forms and the parameters are determined such that the resulting contour map resembles that of the poloidal cross-section for an equilibrium solution of the plasma current density in an ITER-like tokamak (Evangelias & Throumoulopoulos 2016, fig. 4).

Sketch of double-split quadrature. In the numerical integration of the gravitational potential of an axisymmetric object at its internal point, we divide the integration domain into four numbered regions radially and vertically separated at the point. Depicted is a case when the cylindrical coordinates of the internal point are R = 4 and z = 0, which is indicated by a bullet in the figure, for a finite object with a convex cross-section.

Potential contour map of X/peanut-shaped object. Shown is the contour map of the gravitational potential of the object the volume mass density of which is described in Fig. 2. The contours are drawn at every 5 per cent level of the bottom value, which is achieved at the coordinate centre. Omitted is the lower half part where z < 0 since the object is of the x–y plane symmetry. Despite the peculiar feature of the density contours shown in Fig. 2, the potential contours illustrated here are almost elliptical although they are not genuine ellipses.

Potential contour map of power-law disc with hole. Same as Fig. 6 but for the power-law disc the density of which is depicted in Fig. 3. This time, the bottom value is achieved when R ≈ 0.85 and z = 0. If the spherically symmetric potential of a point mass of certain magnitude is superposed, it would result in a realistic model of the gravitational potential of TW Hya.

Potential contour map of power-law disc with hole: close-up. Same as Fig. 7 but focused is the central part. Notice that the disc exists in a limited region, R ≥ 0.2.

Potential contour map of tear-drop-shaped object. Same as Fig. 6 but for the tear-drop-shaped toroid. At this scale, the computed contour map of the gravitational potential looks almost plane symmetric, although the density distribution described in Fig. 4 is not so.

Contour map of kernel function. Illustrated is the contour map of the kernel function of the axisymmetric integral expression of the gravitational potential, g(R′, z′; R, z), when R = 1 and z = 0. The function has a blow-up logarithmic singularity at R′ = R and z′ = z, which is indicated by dense concentric circles around it.

Radial profile of kernel function. Same as Fig. 10 but plotted as functions of R′ for various values of z′ as z′ = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The thick curves are those for z′ = 1 and 5. The curve of z′ = 0 blows up logarithmically at R′ = 1.

Vertical profile of kernel function. Same as Fig. 10 but plotted as functions of z′ for various values of R′ as R′ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.2, 1.4, 1.6, 1.8, 2, 3, 4, 5, 6, 8, 10, 20, and R′ → ∞. The thick curves show the cases when R′ = 0.5, 1, 2, and R′ → ∞. The curve of R′ = 1 blows up logarithmically at z′ = 0. The curve of R′ → ∞ is a constant value of 2π.

Computational cost of vertical integration. Shown is N_eval, the number of function calls required in the numerical integration. Plotted are the results for a line integral representing the gravitational potential of an infinitely thin uniform cylindrical surface of the radius and the half height, a = c = 1. Compared are N_eval as functions of R′, the radius of the location of the vertical line integral, of two methods using the DE quadrature rule with the relative error tolerance δ = 10⁻⁸: (i) the single quadrature without interval splitting, and (ii) the sum of two quadratures for the intervals split at z′ = z. The former method could not obtain the correct integral value when R′ ≈ a.

Computational cost of double-split quadrature. Same as Fig. 13 but for the numerical evaluation of a double integral representing the gravitational potential of a finite uniform cylinder of the radius and the half height as a = c = 1. Compared are two double-split quadrature methods using the DE rules: (i) that of unconditional radial splitting at R′ = R, and (ii) that with the radial splitting only when |R′ − R| < 0.2.

Integration error of gravitational potential: finite uniform spheroid. Shown are the integration errors of the gravitational potential of finite uniform spheroid obtained by the new method. Results are expressed as the base 10 logarithm of the normalized absolute errors of the potential numerically integrated with the relative error tolerance δ = 10⁻¹³. They are plotted as functions of R for several values of z as z = 0, 0.1, 1, and 10 while the semi-axes are fixed as a = 1 and c = 1/2. The errors of different z are overlapped since their distribution is almost the same.

Integration error of gravitational potential of Miyamoto–Nagai model. Same as Fig. 15 but for the Miyamoto–Nagai model (Miyamoto & Nagai 1975) for its model parameters, a_M = 1 and b_M = 1/2.

Computational error of acceleration vector of Miyamoto–Nagai model: radial component. Same as Fig. 16 but for the computational errors of the radial component acceleration.

Computational error of acceleration vector of Miyamoto–Nagai model: vertical component. Same as Fig. 17 but for the vertical component acceleration.

Density contour map of bi-exponential object. Shown is the contour map of the volume mass density of an axisymmetric object expressed as ρ_BE(R, z) ≡ ρ₀ exp (−R/a − |z|/c) when the scalelength parameters are chosen as a = 1 and c = 0.3. The contours are drawn at every 5 per cent level of the peak value. They are rhombi of the aspect ratio, a: c = 10 : 3. Although the contours seem to be localized in a diamond-shaped region as R + |z|/0.3 ≤ 3, the object is infinitely extended in reality.

Density contour map of flared bi-exponential object. Same as Fig. 19 but for an axisymmetric object with the volume mass density profile expressed as ρ_FE(R, z) ≡ ρ₀ exp [−R/a − (|z|/c)/f(R/a)], where f(ξ) is a function describing the normalized vertical scaleheight as f(ξ) = 1 + kξ while the parameters are chosen as a = k = 1 and c = 0.3.

Density contour map of radially exponential and vertically secant-hyperbolic-squared disc. Same as Fig. 19 but when the volume mass density profile is expressed as ρ_ES(R, z) ≡ ρ₀ exp (−R/a) sech²[−|z|/(2c)], where a = 1 and c = 0.3.

Density contour map of radially exponential and vertically Gaussian disc. Same as Fig. 19 but when the volume mass density profile is expressed as ρ_EG(R, z) ≡ ρ₀ exp (−R/a − z²/c²), where a = 1 and c = 0.3.

Density contour map of boxy object. Same as Fig. 19 but when the volume mass density profile is expressed as |$\rho _{\rm BX} (R,z) \equiv \rho _0 \exp \left[ - \sqrt{(R/a)^4 + (z/c)^4} \right]$|⁠, where a = 1 and c = 0.3.

Density profile of Gaussian oval toroid. Same as Fig. E2 but for a toroid with the volume mass density written as ρ_GT(R, z) ≡ ρ₀ exp [−{(R − R_T)/a}² − {(|z|/c)/f(R/a)}²], where f(ξ) ≡ 1 + kξ is a linear flaring function while the parameters are set as R_T = 2, a = 1, c = 0.3, and k = 0.7. We determined the functional form and the parameters to mimic the result of N-body simulation of particle swarm with a toroidal feature (Bannikova, Vakulik & Sergeev 2012, fig. 10).

Potential contour map of bi-exponential object. Same as Fig. 6 but for the bi-exponential object. Its volume mass density is described as ρ_BE(R, z) ≡ ρ₀ exp (−R/a − |z|/c) when a = 1 and c = 0.3. This distribution resembles that of the thick stellar disc of the Milky Way if the unit distance is chosen as 4.0 kpc (Robin et al. 2003, table 3). The contours are drawn at every 5 per cent level of the bottom value. The potential is not analytic on the x–y plane and along the z-axis, although the non-analyticity is hardly visible at this scale.

Potential contour map of bi-exponential object: c = 0.003. Same as Fig. 25 but when c = 0.003. At this scale, observed will be no further difference in the potential contour map when c ≤ 0.003. In this sense, the contour map shown here is practically the same as that of the infinitely thin exponential disc (Freeman 1970). This time, the non-analyticity on the x–y plane is obvious.

Potential contour map of bi-exponential object: c = 0.03. Same as Fig. 25 but when c = 0.03. This approximates the contour map of the gravitational potential of the ISM disc of the Milky Way if the unit distance is chosen as 4.5 kpc (Robin et al. 2003, table 3). Still visible is the non-analyticity on the x–y plane.

Potential contour map of bi-exponential object: c = 3. Same as Fig. 25 but when c = 3.

Potential contour map of bi-exponential object: c = 30. Same as Fig. 25 but when c = 30.

Potential contour map of bi-exponential object: c = 300. Same as Fig. 25 but when c = 300. At this scale, observed will be no further difference in the potential contour map when c ≥ 300.

Potential contour map of flared bi-exponential object. Same as Fig. 25 but for a flared bi-exponential object. Its volume mass density is described as ρ_FE(R, z) ≡ ρ₀ exp [−R/a − (|z|/c)/f(R/a)], where f(ξ) = 1 + kξ. Shown is the result when a = k = 1 and c = 0.3.

Potential contour map of radially exponential and vertically secant-hyperbolic-squared object. Same as Fig. 25 but when the volume mass density profile is given as ρ_ES(R, z) ≡ ρ₀ exp (−R/a) sech²[−|z|/(2c)] when a = 1 and c = 0.3.

Potential contour map of radially exponential and vertically Gaussian object. Same as Fig. 25 but when the volume mass density profile is given as ρ_EG(R, z) ≡ ρ₀ exp (−R/a − z²/c²) when a = 1 and c = 0.3.

Potential contour map of boxy object. Same as Fig. 25 but when the volume mass density profile is expressed as |$\rho _{\rm BX} (R,z) \equiv \rho _0 \exp \left[ - \sqrt{(R/a)^4 + (z/c)^4} \right]$| when a = 1 and c = 0.3.

Potential contour map of differenced Gaussian spheroid. Same as Fig. 25 but when the volume mass density profile is given as |$\rho _{\rm DG} (R,z) \equiv \rho _0 \left[ \exp \left( - s_{\rm out}^2 \right) - \exp \left( - s_{\rm in}^2 \right) \right]$|⁠, where |$s_{\rm out} \equiv \sqrt{R^2/a_{\rm out}^2 + z^2/c_{\rm out}^2}$|⁠, |$s_{\rm in} \equiv \sqrt{R^2/a_{\rm in}^2 + z^2/c_{\rm in}^2}$|⁠, while a_out = 2, c_out = 0.6, a_in = 1, and c_in = 0.3. Notice that the central concentration is significantly reduced.

Potential contour map of differenced modified exponential spheroid. Same as Fig. 35 but when the volume mass density profile is given as ρ_DM(R, z) ≡ ρ₀[exp (−σ_out) − exp (−σ_in)], where |$\sigma _{\rm out} \equiv 2s_{\rm out}^2/ (1 + \sqrt{1+4s_{\rm out}^2})$|⁠, |$\sigma _{\rm in} \equiv 2s_{\rm in}^2/ (1 + \sqrt{1+4s_{\rm in}^2} )$|⁠, while |$s_{\rm out} \equiv \sqrt{R^2/a_{\rm out}^2 + z^2/c_{\rm out}^2}$|⁠, |$s_{\rm in} \equiv \sqrt{R^2/a_{\rm in}^2 + z^2/c_{\rm in}^2}$|⁠, and a_out = 2, c_out = 0.6, a_in = 1, and c_in = 0.3. The central concentration is further reduced.

Potential contour map of Gaussian oval toroid. Same as Fig. 25 but when the volume mass density profile is given as ρ_GT(R, z) ≡ ρ₀ exp [−(R − R_T)²/a² − {(z/c)/f(R/a)}²] while a = 1, c = 0.3, R_T = 2, and k = 0.7.

Rotation curve: bi-exponential object. Shown are the rotation curves of bi-exponential objects the volume mass density of which are expressed as ρ_BE(R, z) ≡ ρ₀ exp (−R/a − |z|/c). Plotted are the circular velocities for various values of c as c = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, and 30 while a is fixed as a = 1. The normalization constant V₀ is arranged such that the total masses of the objects are the same. As a result, all the curves approach to the same functional form, namely that of a point mass, when R → ∞. At this scale, invisible is the departure of the curves when c ≤ 0.01 from the limit of c = 0, namely the rotation curve of the infinitely thin exponential disc described by a cross-product of the modified Bessel functions (Freeman 1970).

Rotation curve: flared bi-exponential object. Same as Fig. 38 but for an object with the volume mass density expressed as ρ_FE(R, z) ≡ ρ₀ exp [−(R/a) − (|z|/c)/f(R/a)], where f(ξ) ≡ 1 + kξ.

Rotation curve of flared bi-exponential object. Same as Fig. 39 but plotted are the curves for various values of k, the flaring factor, while the other parameters are fixed as a = 1, c = 0.3. Compared are the results for k = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5. This time, the nominal density value ρ₀ is set the same. The thick curve is the case of k = 1.

Rotation curve: radially exponential and vertically secant-hyperbolic-squared object. Same as Fig. 38 but for an object with the volume mass density expressed as ρ_ES(R, z) ≡ ρ₀ exp (−R/a)sech²[|z|/(2c)].

Rotation curve: radially exponential and vertically Gaussian object. Same as Fig. 38 but for an object with the volume mass density expressed as ρ(R, z) = ρ₀ exp [−(R/a) − (z/c)²].

Rotation curve: radially exponential and vertically secant-hyperbolic-squared disc. Same as Fig. 38 but for an object with the volume mass density expressed as |$\rho (R,z) = \rho _0 \exp [ - \sqrt{(R/a)^4+(z/c)^4}]$|⁠.

Rotation curve: differenced Gaussian spheroid. Same as Fig. 38 but for an object with the volume mass density expressed as |$\rho _{\Delta {\rm G}} (R,z) \equiv \rho _{\rm G}( s_{\rm out}) - \rho _{\rm G}( s_{\rm in}) = \rho _0[ \exp( - s_{\rm out}^2) - \exp( - s_{\rm in}^2)]$|⁠, where the auxiliary arguments are defined as |$s_{\rm out} \equiv \sqrt{( R/a_{\rm out})^2 + ( z/c_{\rm out})^2}$| and |$s_{\rm in} \equiv \sqrt{( R/a_{\rm in})^2 +( z/c_{\rm in})^2}$| while the constants are set as a_out = 2a, c_out = 2c, a_in = a, and c_in = c.

Rotation curve: differenced modified exponential spheroid. Same as Fig. 44 but for an object with the volume mass density expressed as |$\rho _{\rm DM} (R,z) \equiv \rho _0 [ \exp\lbrace - 2 s_{\rm out}^2 / ( 1 + \sqrt{1 + 4 s_{\rm out}^2}) \rbrace - \exp \lbrace - 2 s_{\rm in}^2 / ( 1 + \sqrt{1 + 4 s_{\rm in}^2}) \rbrace ]$|⁠, where the auxiliary arguments are defined as |$s_{\rm out} \equiv \sqrt{( R/a_{\rm out})^2 +( z/c_{\rm out})^2}$| and |$s_{\rm in} \equiv \sqrt{( R/a_{\rm in})^2 + ( z/c_{\rm in})^2}$| while the constants are set as a_out = 2a, c_out = 2c, a_in = a, and c_in = c.

Rotation curve: Gaussian oval toroid. Same as Fig. 38 but for the Gaussian oval toroid the volume mass density of which is described as ρ_GT(R, z) ≡ ρ₀ exp [−{(R − R_T)/a}² − {(|z|/c)/f(R/a)}²] while f(ξ) ≡ 1 + kξ.

Density distribution dependence of rotation curve. Compared are the rotation curves of eight axisymmetric objects: (i) the bi-exponential object, (ii) the flared bi-exponential object, (iii) the radially exponential and the vertically secant-hyperbolic-function-squared object, (iv) the radially exponential and the vertically Gaussian object, (v) the boxy object, (vi) the differenced modified exponential spheroid, (vii) the differenced Gaussian spheroid, and (viii) the Gaussian oval toroid. Plotted are the rotation curves when a = 1 and c = 0.3 while the velocity V and the radius R are normalized by the maximum speed V_max and the location of maximum speed R_max, respectively. Once this normalization is applied, almost invisible is the difference among the first four objects.

Auxiliary function A(s). Plotted is the curve of an auxiliary function, A(s) ≡ [F(1/2, 1/2; 3/2; s) − 1]/s for the standard interval, |s| ≤ 1. The function is not analytic at s = 1 despite it is finite as A(1) = π/2 − 1 ≈ 0.57.

Potential contour map of uniform oblate spheroid. Plotted are the contours of the gravitational potential of a uniform spheroid with the semi-axes, a = 3 and c = 1. The contours are drawn at every 1 per cent level of the bottom value realized at the coordinate centre. The potential is not analytic on the surface of the spheroid where R²/a² + z²/c² = 1, although the non-analyticity is not obvious at this scale.

Potential contour map of uniform prolate spheroid. Same as Fig. A2 but when the semi-axes are a = 1 and c = 3. Obviously, this figure is different from that of Fig. A2 after the 90° rotation. In fact, clearly visible is the non-analyticity of the potential on the spheroid surface.

Rotation curve: uniform spheroid. Shown are the rotation curves of uniform spheroids of various values of c as c = 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, and 30.0 while a is fixed as a = 1. The normalization constant V₀ is adjusted such that the total mass of the object becomes the same. As a result, the limit curve of R → ∞ becomes the same, namely that of a point mass. At this scale, invisible is the departure of the curves when c ≤ 0.03 from the limit of c = 0. All the curves have a kink at the surface of the spheroid, namely when R = a.

Potential contour map of finite uniform cylinder. Same as Fig. 25 but for a finite uniform cylinder of the radius a = 1 and the half height c = 1. Namely, the object occupies the volume defined as 0 ≤ R ≤ a and 0 ≤ |z| ≤ c. The contours are drawn at every 2 per cent level of the bottom value. Easily visible is the non-analyticity of the potential on the surface of the cylinder, R = a and/or |z| = c. The map is not symmetric with the 90° rotation.

Potential contour map of finite uniform cylinder, a = 3 and c = 1. Same as Fig. D1 but when a = 3 and c = 1.

Potential contour map of finite uniform cylinder, a = 1 and c = 3. Same as Fig. D1 but when a = 1 and c = 3. Notice that this figure is not the same as Fig. D2 even after the 90° rotation.

Potential contour map of finite uniform rhombic spindle. Same as Fig. D1 but for a finite uniform rhombic spindle of the radial and vertical semi-axes, a = 3 and c = 1, respectively. Namely, the object occupies the volume defined as 0 ≤ R/a + |z|/c ≤ 1.

Potential contour map of finite uniform flared cylinder. Same as Fig. D1 but for a finite uniform flared cylinder of the radial and vertical semi-axes, a = 3 and c = 1, respectively. Namely, the object occupies the volume defined as 0 ≤ R/a ≤ |z|/c ≤ 1.

Potential contour map of finite uniform bipolar cone. Same as Fig. D1 but for a finite uniform bipolar cone of the radial and vertical semi-axes, a = 1 and c = 3, respectively. Namely, the object occupies the volume defined as 0 ≤ |z|/c ≤ R/a ≤ 1.

Potential contour map of pair of finite uniform cylinders. Same as Fig. D1 but for a pair of cylinders with the radius and the half height as a = c = 1 and separated by Δz = 2. Namely, the objects occupy the volume defined as 0 ≤ R ≤ a and 0 ≤ |z ± Δz| ≤ c.

Potential contour map of finite uniform square toroid. Same as Fig. D1 but for a finite uniform square toroid of the inner and outer radii, R_in = 1 and R_out = 3. Namely, the object occupies the volume defined as R_in ≤ R ≤ R_out and 0 ≤ |z| ≤ (R_out − R_in)/2.

Potential contour map of finite circular uniform toroid. Same as Fig. D1 but for a finite uniform toroid of the inner and outer radii, R_in = 1 and R_out = 3, respectively. Namely, the object occupies the volume defined as 0 ≤ [R − (R_in + R_out)/2]² + z² ≤ [(R_out − R_in)/2]². Thus, centre of the cross-section of the toroid is at R = 2 and z = 0. Meanwhile, the location of the potential maximum is at R ≈ 1.45 and z = 0.

Potential contour map of finite uniform hemisphere. Same as Fig. D1 but for a finite uniform hemisphere of the radius a = 1. Namely, the object occupies the volume defined as 0 ≤ R² + z² ≤ a² and 0 ≤ z.

Potential contour map of finite uniform cone. Same as Fig. D1 but for a finite uniform cone of the radius a = 1 and the height c = 2. Namely, the object occupies the volume defined as 0 ≤ z/c ≤ R/a ≤ 1.

Rotation curve: finite uniform cylinder. Shown are the rotation curves of finite uniform cylinders of the radius a and the half height c. Depicted are the results for various values of c as c = 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, and 30.0 while a is fixed as a = 1. The normalization constant V₀ is adjusted such that the total mass of the object is the same. As a result, the limit curve of R → ∞ becomes the same, namely that of a point mass. At this scale, invisible is the departure of the curves when c ≤ 0.03 from the limit of c = 0, namely that of the infinitely thin uniform disc (Fukushima 2016, fig. 5). The curves have a kink at the surface of the cylinder, namely when R = a. The kink approaches to a logarithmic blow-up singularity when c → 0.

Rotation curve: finite uniform rhombic spindle. Same as Fig. D12 but for finite uniform rhombic spindles.

Rotation curve: finite uniform flared cylinder. Same as Fig. D12 but for finite uniform flared cylinders.

Rotation curve: finite uniform bipolar cone. Same as Fig. D12 but for finite uniform bipolar cones.

Rotation curve: finite uniform double cylinder. Same as Fig. D7 but for the rotation curve for several values of Δz as Δz = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.2, 2.4, 2.6, 2.8, and 3.0 while the radius and the half height are fixed as a = c = 1.

Rotation curve: finite uniform circular toroid. Same as Fig. 38 but for a uniform circular toroid for several values of R_in as R_in = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, and 1.9, and the outer radius is fixed as R_out = 2. Meanwhile, the half height is arranged as c ≡ (R_out + R_in)/2 such that the cross-section is square-shaped. All the curves have a kink at R = R_out.

Rotation curve: finite uniform circular toroid. Same as Fig. 38 but for a uniform circular toroid for several values of R_in as R_in = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, and 1.9, while the outer radius is fixed as R_out = 2. All the curves have a kink at R = R_out.

Density profile of some spheroids. Plotted are the volume mass density of some spheroids: (i) the Gaussian spheroid, ρ_G(s) ≡ ρ₀ exp (−s²), where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠, (ii) the exponential spheroid, ρ_X(s) ≡ ρ₀ exp (−s), (iii) the modified exponential spheroid, |$\rho _{\rm M} (s) \equiv \rho _0 \exp \left[ - s^2 \left/ \left( 1 + \sqrt{1+s^2} \right) \right] \right.$|⁠, and (iv) the Burkert spheroid (Burkert 1995), ρ_B(s) ≡ ρ₀(1 + s)⁻¹(1 + s²)⁻¹. The thick curve shows the Gaussian spheroid. The exponential and Burkert spheroids have a cusp at the coordinate centre.

Density profile of indexed Plummer spheroids. Same as Fig. E1 but for some indexed Plummer spheroids, ρ_Pj(s) ≡ ρ₀(1 + s²)^−j/2, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠. Here (i) the case j = 2 is the modified isothermal model (King 1962), (ii) the case j = 3 is the modified Hubble model (Rood et al. 1972), (iii) the case j = 4 is the perfect spheroid (de Zeeuw 1985a,b), and (iv) the case j = 5 is the original Plummer model (Plummer 1911).

Density profile of double power-law spheroids. Same as Fig. E2 but plotted are double logarithmic graphs for some spheroids with the double power-law density profile: (i) the Jaffe spheroid (Jaffe 1983), ρ_J(s) ≡ ρ₀s⁻²(1 + s)⁻², where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠, (ii) the Hernquist spheroid (Hernquist 1990), ρ_H(s) ≡ ρ₀s⁻¹(1 + s)⁻³, and (iii) the Navarro–Frenk–White (NFW) spheroid (Navarro, Frenk & White 1997), ρ_N(s) ≡ ρ₀s⁻¹(1 + s)⁻². The result of the Jaffe spheroid is depicted in a broken curve while those of the NFW and Hernquist spheroids are shown by a thick and a thin solid curve.

Density profile of Sérsic spheroids. Same as Fig. E2 but plotted are the single logarithmic graphs for the spheroids approximately resulting the Sérsic law of the surface intensity profile (Sérsic 1963, 1968). Here the volume mass density is written as |$\rho _{{\rm S}n} (s) \equiv \rho _0 s^{-p_n} \exp ( - s^{1/n} )$|⁠, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| while the power index p_n are approximated (Lima Neto, Gerbal & Márquez 1999) as p_n = 1 − 0.6097n + 0.054 63n². Compared are the curves of (i) n = 1 drawn by a thick line, (ii) n = 2 by a dotted line, and (iii) n = 4 by a thin line. The case of the index n = 4 corresponds to the famous de Vaucouleurs’ law (de Vaucouleurs 1948).

Density profile of Einasto spheroids. Same as Fig. E2 but plotted are modified logarithmic graphs for the Einasto spheroids. Their volume mass density profile is expressed as ρ_Ti(s) ≡ ρ₀ exp (−d_is^1/i), where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| while the coefficient d_i are approximated (Merritt et al. 2006, equation 24) as d_i = 3i − 1/3 + 0.0079/i. Compared are the curves of some indices as n = 1/2, 1, 2, 4, and 7. Except the scaling constants, the curves of n = 1/2 and 1 are the same as those of the Gaussian and exponential spheroids, respectively.

Potential contour map of Gaussian spheroid. Shown is the contour map of the gravitational potential of the Gaussian spheroid defined as ρ_G(s) ≡ ρ₀ exp (−s²) while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3. The contours are drawn at every 5 per cent level of the bottom value realized at the coordinate origin. Despite their elliptical outlook, the contours are not genuine ellipses.

Potential contour map of exponential spheroid. Same as Fig. E6 but for the exponential spheroid defined as ρ_X(s) ≡ ρ₀ exp (−s) while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of modified exponential spheroid. Same as Fig. E6 but for the modified exponential spheroid (Robin et al. 2003) defined as |$\rho _{\rm M} (s) \equiv \rho _0 \exp [ - s^2/ \sqrt{1 + \sqrt{1+s^2}} ]$| while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of Burkert spheroid. Same as Fig. E6 but for the Burkert spheroid (Burkert 1995) defined as ρ_B(s) ≡ ρ₀(1 + s)⁻¹(1 + s²)⁻¹ while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of modified isothermal spheroid. Same as Fig. E6 but for a modified isothermal spheroid (King 1962) defined as ρ_P2(s) ≡ ρ₀(1 + s²)⁻¹ while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of modified Hubble spheroid. Same as Fig. E6 but for the modified Hubble spheroid (Binney & Tremaine 2008) defined as ρ_P3(s) ≡ ρ₀(1 + s²)^−3/2 while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of perfect spheroid. Same as Fig. E6 but for the perfect spheroid (de Zeeuw 1985a,b) defined as ρ_P4(s) ≡ ρ₀(1 + s²)⁻² while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of Plummer spheroid. Same as Fig. E6 but for the Plummer spheroid (Plummer 1911) defined as ρ_P5(s) ≡ ρ₀(1 + s²)^−5/2 while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of Jaffe spheroid. Same as Fig. E6 but for a Jaffet spheroid (Jaffe 1983) defined as ρ_J(s) ≡ ρ₀s⁻²(1 + s)⁻² while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of Hernquist spheroid. Same as Fig. E6 but for a Hernquist spheroid (Hernquist 1990) defined as ρ_H(s) ≡ ρ₀s⁻¹(1 + s)⁻³ while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of NFW spheroid. Same as Fig. E6 but for a Navarro–Frenk–White (NFW) spheroid (Navarro et al. 1997) defined as ρ_N(s) ≡ ρ₀s⁻¹(1 + s)⁻² while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| when a = 1 and c = 0.3.

Potential contour map of Sérsic spheroid, n = 1. Same as Fig. E6 but for a Sérsic spheroid (Sérsic 1963, 1968) of index n = 1 defined as |$\rho _{{\rm S}1} (s) \equiv \rho _0 s^{-p_1} \exp \left( - s \right)$| while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and p₁ ≈ 0.444 930 00 when a = 1 and c = 0.3.

Potential contour map of Sérsic spheroid, n = 2. Same as Fig. E6 but for a Sérsic spheroid (Sérsic 1963, 1968) of index n = 2 defined as |$\rho _{{\rm S}2} (s) \equiv \rho _0 s^{-p_2} \exp \left( - \sqrt{s} \right)$| while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and p₂ ≈ 0.708 807 50 when a = 1 and c = 0.3.

Potential contour map of de Vaucouleurs spheroid. Same as Fig. E6 but for a de Vaucouleurs spheroid (de Vaucouleurs 1948), namely Sérsic spheroid of index n = 4 defined as |$\rho _{{\rm S}4} (s) \equiv \rho _0 s^{-p_4} \exp \left( - s^{1/4} \right)$| while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and p₄ ≈ 0.850 989 37 when a = 1 and c = 0.3.

Potential contour map of Einasto spheroid, i = 2. Same as Fig. E6 but for an Einasto spheroid (Einasto 1965) of index 2 defined as |$\rho _{{\rm E}2} (s) \equiv \rho _0 \exp \left( - d_2 \sqrt{s} \right)$| while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and d₂ ≈ 5.706 1667 when a = 1 and c = 0.3.

Potential contour map of Einasto spheroid, i = 4. Same as Fig. E6 but for an Einasto spheroid (Einasto 1965) of index 4 defined as ρ_E4(s) ≡ ρ₀ exp (−d₄s^1/4) while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and d₄ ≈ 11.686 417 when a = 1 and c = 0.3.

Potential contour map of Einasto spheroid, i = 7. Same as Fig. E6 but for an Einasto spheroid (Einasto 1965) of index 7 defined as ρ_E7(s) ≡ ρ₀ exp (−d₇s^1/7) while |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| and d₇ ≈ 20.677 952 when a = 1 and c = 0.3.

Rotation curve of Gaussian spheroid. Shown are the rotation curves of a group of spheroids with the volume mass density expressed as ρ_G(s) ≡ ρ₀ exp (−s²), where |$s \equiv \sqrt{(R/a)^2 + (z/c)^2}$|⁠. Plotted are the curves for various values of c as c = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5 while a is set as a = 1 and the nominal density value, ρ₀, is fixed. If the total mass is fixed instead, V₀ is adjusted such that the asymptotic forms of all the curves become the same. The thick curve represents the spherically symmetric case, c = a, which is analytically computable by means of the error function.

Rotation curve of modified exponential spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_X(s) ≡ ρ₀ exp (−s), where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠.

Rotation curve of modified exponential spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as |$\rho _{\rm M} (s) \equiv \rho _0 \exp [- 2 s^2/( 1 + \sqrt{1+4s^2} ) ]$|⁠, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠.

Rotation curve of Burkert spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_B(s) ≡ ρ₀(1 + s)⁻¹(1 + s²)⁻¹, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Burkert 1995).

Rotation curve of modified isothermal spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_P2(s) ≡ ρ₀(1 + s²)⁻¹, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (King 1962).

Rotation curve of modified Hubble spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_P3(s) ≡ ρ₀(1 + s²)^−3/2, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Binney & Tremaine 2008).

Rotation curve of perfect spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_P4(s) ≡ ρ₀(1 + s²)⁻², where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (de Zeeuw 1985a,b).

Rotation curve of Plummer spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_P5(s) ≡ ρ₀(1 + s²)^−5/2, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Plummer 1911).

Rotation curve of Jaffe spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_J(s) ≡ ρ₀s⁻²(1 + s)⁻², where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Jaffe 1983). Notice that the circular speed approaches to a finite value when R → 0. This is unphysical.

Rotation curve of Hernquist spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_H(s) ≡ ρ₀s⁻¹(1 + s)⁻³, where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Hernquist 1990).

Rotation curve of Navarro–Frenk–White (NFW) spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_NFW(R, z) ≡ ρ₀s⁻¹(1 + s)⁻², where |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Navarro et al. 1997).

Rotation curve of Sérsic spheroid, n = 1. Same as Fig. E23 but for a spheroid with the volume mass density expressed as |$\rho _{{\rm S}1} (s) \equiv \rho _0 s^{-p_1} \exp (-s)$|⁠, where p₁ ≈ 0.444 930 00 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠. This density profile approximately reproduces the surface brightness profile following the Sérsic law of index n = 1 (Sérsic 1963, 1968).

Rotation curve of Sérsic spheroid, n = 2. Same as Fig. E23 but for a spheroid with the volume mass density expressed as |$\rho _{{\rm S}2} (s) \equiv \rho _0 s^{-p_2} \exp \left( - \sqrt{s} \right)$|⁠, where p₂ ≈ 0.708 807 50 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠. Enlarged is the range of radial distance so as to cover the feature of rise and fall.

Rotation curve of de Vaucouleurs spheroid. Same as Fig. E23 but for a spheroid with the volume mass density expressed as |$\rho _{{\rm S}4} (s) \equiv \rho _0 s^{-p_4} \exp \left( - s^{1/4} \right)$|⁠, where p₄ ≈ 0.850 989 37 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠. This time, further extended is the range of radial distance so as to cover the feature of rise and fall. If the object is spherically symmetric such that a = c, the volume density profile approximately reproduces the surface brightness profile following the de Vaucouleurs’ law (de Vaucouleurs 1948).

Rotation curve of Einasto spheroid, i = 2. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_E2(s) ≡ ρ₀ exp (−d₂ s), where d₂ ≈ 5.706 1667 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$|⁠.

Rotation curve of Einasto spheroid, i = 4. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_E4(s) ≡ ρ₀ exp (−d₄s^1/4), where d₄ ≈ 11.686 417 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Einasto 1965). This time, the radial argument range is diminished so as to emphasize the initial rising feature.

Rotation curve of Einasto spheroid, i = 7. Same as Fig. E23 but for a spheroid with the volume mass density expressed as ρ_E7(s) ≡ ρ₀ exp (−d₇s^1/7), where d₇ ≈ 20.677 952 and |$s \equiv \sqrt{(R/a)^2+(z/c)^2}$| (Einasto 1965). Again, the radial argument range is diminished so as to emphasize the initial rising feature.

Spheroidal law dependence of rotation curve. Same as Fig. 47 but compared are the rotation curves of most of the spheroids when a = 1 and c = 0.3. The exceptions are (i) the Jaffe spheroid where the curve reaches the maximum at the coordinate origin, and (ii) the modified isothermal spheroid where the curve increases monotonically and no maximum value is attained. Among the curves shown here, that of the Einasto spheroid of the index i = 7 increases most quickly and decreases most gradually. Meanwhile, that of the Gaussian spheroid grows most slowly and decays most rapidly. See Table E1 for the detailed comparison.