The accuracy of the gravitational potential was determined by comparison of our numerical solution

Transcription

1 the results of three key accuracy tests. 48 A6.1. Gravitational Potential The accuracy of the gravitational potential was determined by comparison of our numerical solution with the analytical result for a sphere of radius a < R centered at the origin. On the outer boundary, the potential is that due to a point source at the origin. The numerical result has a relative error of a fraction of a percent for the potential, and a few percent for the field components on a grid, for both uniform and nonuniform cells. The largest errors occur at the surface of the sphere, reflecting the fact that averaging the density over cells that straddle the edge of the sphere is the dominant contribution to the inaccuracy of the numerical solution. In practice, such sharp changes in the values of functions are either resolved by the adaptive grid or smeared over several zones by viscosity forces; therefore, we expect truncation errors in the gravitational field components to be considerably less than a few percent for most MHD problems. A6.2. Free Fall A useful test problem for which there is a known analytical solution is the gravitational collapse of a spherically symmetric, nonuniform, cold cloud of infinite size starting from rest (e. g., see Spitzer 1978). The initial state has a Gaussian density profile. Boundary values for the density, velocity, and gravitational potential are obtained from the analytical solution by numerical quadrature at the mean positions of cells which are the images of those lying along the outer boundary of the computational region. Because the density becomes sharply peaked, this is a severe test of the accuracy of the spatial differencing, interpolation, gravitational force, advection for a smooth flow, and the adaptive grid, but not of the pressure or magnetic forces. After a density enhancement of five orders of magnitude on a grid, the largest relative error in any dependent variable is about 10%, which is in the density of the cell at the origin. The lengths of the sides of this cell are smaller by a factor of 50 from their values in the initial state, which had a grid that was nearly uniform. After a density enhancement of six orders of magnitude, the largest relative error, which is again in the density at the origin, has

2 increased to 20%. The density contours remain nearly spherical. Since the density profile should be 49 less sharply peaked at the origin when pressure and magnetic forces are included in a more realistic problem, truncation errors are expected to be significantly lower in practice than the values for these test runs, provided the mesh points are positioned properly. A6.3. Collapse Under Flux-Freezing Under the assumption that the magnetic flux is frozen in the neutrals, we have followed the evolution of a model cloud whose innermost flux tubes are magnetically and thermally supercritical. The initial, spherically symmetric density distribution is given by n(r, z) = (n 0 n S ) cos 4 π 2R r 2 + z 2 + n S, r 2 + z 2 R, (A51a) = n S, r 2 + z 2 > R, (A51b) where n 0 and n S have values of and cm 3, respectively, and R = cm. The 60 M O object has B = B z = 22.8 µg, T = 10 K, and α 0 = 5. The central mass-to-flux ratio is greater than the total value by a factor of 2.8 and is 1.3 times the supercritical value (see eq. [8]). By the end of a test run on a grid, the central density has reached cm 3 at t f = yr (= 2.6 initial free-fall times). The solid curve in Figure 2 shows the central magnetic field strength B c versus the central density n c. The dashed curve gives the slope κ d(ln B c )/d(ln n c ). At the start of the run, the fragment begins to contract spherically and κ 2/3 because the uniform magnetic field exerts no forces. The field strength soon increases due to the deformation of the field lines as the cloud contracts. The magnetic force increases sufficiently to reverse the radial component of the velocity, but the density continues to increase as matter flows down along field lines. The field weakens as it pushes the matter outward, but then increases monotonically once the material resumes its inward motion. After the density is enhanced beyond an order of magnitude, force balance between pressure and gravity is established along field lines, and κ oscillates about the expected value of 1/2 (see Mouschovias 1976b). Figure 3a shows field lines and density contours at t f for the entire fragment. Along the axis, the density decreases from

3 50 its central value to cm 3 at z = Z. Matter in the supercritical flux tubes has formed a very thin disk inside the radius 0.2 R, while outside 0.8 R, in the magnetically supported envelope, the field lines have barely moved. The differential mass-to-flux ratio at the axis, obtained by evaluating the integral in equation (8) numerically, has changed by less than 0.2%, which is a negligible amount compared to the change expected when ambipolar diffusion is included. Note that exact conservation of the differential mass-to-flux ratio is not guaranteed by our numerical scheme because the density and flux are advected separately. The filtered and unfiltered meshes at t f are displayed in Figures 3b and 3c, respectively. Note that in the lower left quadrant of the two plots, adjacent zone sides are much more nearly orthogonal in the filtered mesh than in the unfiltered one. Truncation errors in spatial derivatives are smallest on an orthogonal mesh, even though the order of the error terms is independent of the zone shape. Figures 4a, 4b, and 4c show the field lines superposed on the density contours, the filtered mesh, and the unfiltered mesh, respectively, at t f for the innermost 2% of the fragment. The ratio of equatorial and polar radii of the innermost isodensity contours exceeds 10. The unfiltered mesh is extremely distorted, with small angles between adjacent zone sides and very abrupt changes in lengths, whereas the filtered mesh varies quite smoothly. The height and width of the innermost filtered mesh zone have decreased from the values for a uniform mesh by factors of 180 and 45, respectively. Along the axis, there are roughly three zones resolving any region over which the density varies by an order of magnitude. The numerical parameters for the grid (discussed in A5.1 and A5.2) had the following values throughout the calculation: w u = 5, w ρ = 4, w v = 1, w q = 0, w m = 1, m = 2, w n = , n = 2, w p = 1, p = 2, w a = w b = w e = w h = 30, l = 2.

4 REFERENCES 51 Black, D. C., & Scott, E. H. 1982, ApJ, 263, 696 Bonner, W. B. 1956, MNRAS, 116, 351 Ebert, R. 1955, Z. Astrophys., 37, , Z. Astrophys., 42, 263 Elmegreen, B. G. 1979, ApJ, 232, , in Light on Dark Matter, ed. F. P. Israel (Dordrecht: Reidel), 265 Evans, C. R., & Hawley, J. F. 1988, ApJ, 332, 659 Falgarone, E., and Puget, J. L. 1985, A&A, 142, 157 Fiedler, R. A. 1990, Ph. D. thesis, University of Illinois at Urbana-Champaign Fiedler, R. A., & Mouschovias, T. Ch. 1992, ApJ, in preparation Gear, C. W. 1971, Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs: Prentice-Hall) Hindmarsh, A. C. 1983, in Scientific Computing, ed. R. S. Stepleman et al., Vol. 1 (IMACS Transactions on Scientific Computation) (Amsterdam: North Holland), 55 Kulsrud, R., & Pearce, W. P. 1969, ApJ, 156, 445 Lizano, S., & Shu, F. H. 1989, ApJ, 342, 834 McDaniel, E. W., & Mason, E. A. 1973, The Mobility and Diffusion of Ions in Gases (New York: Wiley) Mestel, L., & Spitzer, L. Jr. 1956, MNRAS, 116, 504 Mouschovias, T. Ch. 1976a, ApJ, 206, b, ApJ, 207, , ApJ, 211, , in Protostars and Planets, ed. T. Gehrels (Tucson: Univ. of Arizona Press), , ApJ, 228, , in Solar and Stellar Magnetic Fields: Origins and Coronal Effects, ed. J. O. Stenflo (Dordrecht: Reidel), 479

5 . 1987a, in Physical Processes in Interstellar Clouds, ed. G. E. Morfill & M. Scholer 52 (Dordrecht: Reidel), b, in Physical Processes in Interstellar Clouds, ed. G. E. Morfill & M. Scholer (Dordrecht: Reidel), , in The Physics and Chemistry of Interstellar Molecular Clouds, ed. G. Winnewisser & J. T. Armstrong (Berlin: Springer), a, in Galactic and Intergalactic Magnetic Fields, ed. R. Beck, P. P. Kronberg, & R. Wielebinski (Dordrecht: Reidel), b, in Physical Processes in Fragmentation and Star Formation, ed. R. Capuzzo- Dolcetta, C. Chiosi, & A. di Fazio (Dordrecht: Kluwer), , ApJ, 373, 169 Mouschovias, T. Ch., Ciolek, G., & Morton, S. A. 1992, ApJ, in preparation Mouschovias, T. Ch., & Morton, S. A. 1985, ApJ, 298, , ApJ, 371, a, ApJ, in press. 1992b, ApJ, in press Mouschovias, T. Ch., Morton, S. A., & Ciolek, G. 1992, ApJ, in preparation Mouschovias, T. Ch., & Paleologou, E. V. 1986, ApJ, 308, 781 Mouschovias, T. Ch., Paleologou, E. V., & Fiedler, R. A. 1985, ApJ, 291, 772 Mouschovias, T. Ch., & Spitzer, L., Jr. 1976, ApJ, 210, 326 Nakano, T. 1979, PASJ, 31, 697 Nakano, T., & Tademaru, T. 1972, ApJ, 173, 87 Paleologou, E. V., & Mouschovias, T. Ch. 1983, ApJ, 275, 838 Scott, E. H., & Black, D. C. 1980, ApJ, 239, 166 Shu, F. H. 1983, ApJ, 273, 202 Spitzer, L., Jr. 1968, Diffuse Matter in Space (New York: Interscience). 1978, Physical Processes in the Interstellar Medium (New York: Wiley-Interscience)

6 Tomisaka, K., Ikeuchi, S., & Nakamura, T. 1988, ApJ, 335, , ApJ, 362, 202 Tscharnuter, W. M., & Winkler, K.-H. A. 1979, Comput. Phys. Comm., 18, 171 van Leer, B. 1979, J. Comput. Phys., 32, 101

7 CAPTIONS TO FIGURES 54 FIG. 1. A portion of the nonuniform, nonorthogonal grid. The cell indices are indicated in parentheses. The points labeled "α", "β", "γ", and "δ" are grid points, while points a, c, and e are at the volume-averaged coordinates of their respective cells. The points b, T, and d are at the surfaceaveraged coordinates of the left, top, and right sides of cell (i, j). FIG. 2. The central magnetic field strength B c vs. the central density n c (scale on left, solid curve), and the slope κ d(ln B c )/d(ln n c ) (scale on right, dashed curve). Initially, κ 2/3, the value for spherical contraction. After force balance has been established between pressure and gravity along magnetic field lines, κ oscillates about the expected value of one half. FIG. 3. (a) Density contours and magnetic field lines for the entire fragment. Three contours are shown per order of magnitude of variation in the density, with the uppermost contour corresponding to the lowest value, 300 cm 3. The next lowest value is 600, and then 1000 cm 3. This pattern is repeated up to n c = cm 3. Field lines beyond the radius 0.8 R have hardly moved. (b) Filtered mesh, and (c) unfiltered mesh for the entire computational region. In the lower left quadrant, adjacent zone sides in the filtered mesh are much more nearly orthogonal (allowing more accurate spatial differencing) than they are in the unfiltered mesh. FIG. 4. (a) Density contours and magnetic field lines for the innermost 2% of the fragment. Several contours are labeled by the corresponding value of the density. Typically, the ratio of equatorial and polar radii exceeds 10. Along the z-axis, the density decreases by more than four orders of magnitude as z increases from 0 to Z. Structures smaller than zone sizes (see b) in some contours are an artifact of the contouring algorithm, which was not designed for the widely varying spatial resolution of our mesh. (b) Filtered mesh, and (c) unfiltered mesh for the innermost 2% of the computational region. The filtered mesh varies smoothly, while the unfiltered mesh is extremely distorted, with small angles between zone sides, abrupt variations of lengths, and some tangling. The

8 filtered mesh provides roughly three well-placed cells per order of magnitude of density variation, 55 whereas the unfiltered mesh has far more zones inside the innermost isodensity contour.

9 POSTAL ADDRESSES: Robert A. Fiedler National Center for Supercomputing Applications University of Illinois 5325 Beckman Institute 405 N. Mathews Avenue Urbana, IL Telemachos Ch. Mouschovias Department of Astronomy University of Illinois 1002 W. Green St. Urbana, IL 61801