Astrophyzix SolarForm
Real‑time solar system formation in your browser.
SolarForm is a real‑time, browser‑native simulation of solar system formation, modelling nebular collapse, disk evolution, planetesimal growth, and N‑body orbital dynamics. Using physically grounded equations and provenance‑linked parameters, the module visualises how dust grains become worlds through accretion, migration, and gravitational interaction.
t = 0.000 Myr
NEBULA COLLAPSE
Protostar
Planetesimal
Proto-Planet
Rocky Planet
Ice Body
Gas Giant
Simulation Parameters
Nebula Mass 1.0 M_sun
Angular Momentum 1.0 x
Dust/Gas Ratio 0.01
Snowline (AU) 2.7 AU
N Bodies 80
Sim Speed 1.0 x
Governing Equations
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1. Gravitational Force (Newton / General Form)
F_ij = -G * m_i * m_j / |r_ij|^2 * r_hat_ij
G (SI) = 6.674e-11 N m^2 kg^-2
G (sim) = 4 * pi^2 AU^3 yr^-2 M_sun^-1 [used in code]
r_ij = r_j - r_i (separation vector, AU)
r_hat_ij = r_ij / |r_ij| (unit vector)
Softened: |r_ij|_s = sqrt(|r_ij|^2 + eps^2), eps = 0.005 AU
Pairwise Newtonian gravity between all bodies. The code uses G = 4*pi^2 in AU/yr/M_sun units (exact by Kepler's third law). A fixed softening length of eps = 0.005 AU prevents force singularities at close separations.
2. Velocity Verlet Integration (Symplectic)
r(t+dt) = r(t) + v(t)*dt + 0.5*a(t)*dt^2
a(t+dt) = F(r(t+dt)) / m
v(t+dt) = v(t) + 0.5*(a(t) + a(t+dt))*dt
Second-order symplectic integrator. Conserves energy far better than simple Euler and is well-suited to long orbital integrations. The code uses a fixed base time step DT_BASE = 0.0002 yr per substep. The number of substeps per animation frame scales with the speed multiplier (steps = speed * 4), so faster simulation speeds increase temporal resolution per frame rather than enlarging dt. At 1 AU this keeps orbital phase error below 0.2 percent per orbit.
3. Protoplanetary Disk Surface Density (MMSN)
Sigma(r) = Sigma_0 * (r / r_0)^-3/2
Sigma_0 = 1700 g cm^-2 (inner disk, MMSN)
r_0 = 1 AU
Valid for: 0.35 AU to ~36 AU
Minimum Mass Solar Nebula (MMSN) surface density profile derived by Hayashi (1981). The r^(-3/2) power law is inferred from current planetary masses redistributed as a smooth disk.
4. Keplerian Orbital Velocity
v_K(r) = sqrt(G * M_star / r)
G = 4 * pi^2 AU^3 yr^-2 M_sun^-1
M_star = nebula mass * f_star (f_star = 0.999, in M_sun)
r = radial distance from protostar (AU)
v_K = result in AU yr^-1
Initial tangential velocity for each planetesimal set to circular Keplerian speed, then scaled by the angular momentum multiplier. A small eccentricity perturbation (2-6 percent) is added. Perturbations from N-body interactions then drive eccentricity growth and eventual accretion.
5. Accretion / Inelastic Collision
If |r_ij| less than f_grav * (r_phys_i + r_phys_j):
m_new = m_i + m_j
p_new = m_i*v_i + m_j*v_j (momentum conserved)
v_new = p_new / m_new
r_new = (m_i*r_i + m_j*r_j) / m_new (COM)
f_grav = 8 (gravitational focusing enhancement)
r_phys = (3m / 4*pi*rho)^(1/3) [AU]
rho = 3000 kg m^-3 (rocky, planetesimal, proto-planet inside snowline)
= 1000 kg m^-3 (ice, gas giant, bodies beyond snowline)
Perfectly inelastic sticking collision. Physical radii are computed from bulk density; the factor f_grav = 8 approximates gravitational focusing (Safronov 1969), which enlarges the effective collision cross-section beyond the geometric radius at low relative velocities typical of near-circular disk orbits. Post-merger type is classified by mass and snowline position using three thresholds: Rocky Planet at 5e-7 M_sun (~0.17 Earth masses, inside snowline), Proto-Planet at 5e-6 M_sun (~1.5 Earth masses), and Gas Giant at 3e-4 M_sun (~100 Earth masses).
6. Jeans Instability / Nebula Fragmentation
M_J = (5 k_B T / G m_H2 mu)^(3/2) * (3 / 4*pi*rho)^(1/2)
k_B = 1.381e-23 J K^-1
m_H2 = 3.347e-27 kg
mu = mean molecular weight (~2.3 for solar)
T = cloud temperature (~10-30 K)
rho = cloud density
Jeans mass sets the characteristic fragmentation scale of the molecular cloud. Only clumps exceeding M_J collapse gravitationally. This equation provides the physical motivation for the simulation starting condition (a collapsed disk) but is not evaluated numerically at runtime; the initial disk mass is set directly via the Nebula Mass and Dust/Gas Ratio parameters.
7. Angular Momentum Conservation
L = sum_i( m_i * (r_i cross v_i) ) = constant
For a thin Keplerian disk:
L_disk = M_disk * sqrt(G * M_star * a)
a = semi-major axis of equivalent circular orbit
Angular momentum of the disk particle subsystem is approximately conserved; the protostar is held fixed at the origin (infinite effective mass approximation), consistent with M_star comprising 99.9 percent of total system mass. The angular momentum slider scales all initial tangential velocities by a uniform multiplier, controlling the initial orbital radii and the degree of disk flattening.
8. Snowline Temperature Profile
T(r) = 280 * (L_star / L_sun)^(1/4) * (r / 1 AU)^(-1/2) [K]
Snowline at T = 170 K:
r_snow = (280 / 170)^2 * (L_star/L_sun)^(1/2) AU
~ 2.7 AU for L_star = L_sun
Passive irradiation disk temperature profile (Kenyon and Hartmann 1987). Beyond the snowline, water ice condenses, increasing solid surface density by ~4x and enabling gas giant core formation by streaming instability.
Scientific Provenance and References
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[1] MMSN MODEL
Hayashi, C. (1981). Structure of the solar nebula, growth and decay of magnetic fields and effects of magnetic and turbulent viscosities on the nebula. Progress of Theoretical Physics Supplement, 70, 35-53. doi:10.1143/PTPS.70.35
[2] DISK TEMPERATURE PROFILE
Kenyon, S. J., and Hartmann, L. (1987). Spectral energy distributions of T Tauri stars - Disk flaring and limits on accretion. Astrophysical Journal, 323, 714-733.
[3] VELOCITY VERLET INTEGRATION
Verlet, L. (1967). Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical Review, 159(1), 98-103. doi:10.1103/PhysRev.159.98
[4] JEANS INSTABILITY
Jeans, J. H. (1902). The stability of a spherical nebula. Philosophical Transactions of the Royal Society A, 199, 1-53. doi:10.1098/rsta.1902.0012
[5] PLANETESIMAL ACCRETION
Wetherill, G. W., and Stewart, G. R. (1989). Accumulation of a swarm of small planetesimals. Icarus, 77(2), 330-357. doi:10.1016/0019-1035(89)90093-6
[6] SNOWLINE AND GAS GIANT FORMATION
Johansen, A., et al. (2007). Rapid planetesimal formation in turbulent circumstellar disks. Nature, 448, 1022-1025. doi:10.1038/nature06086
[7] N-BODY SOFTENING
Aarseth, S. J. (2003). Gravitational N-Body Simulations: Tools and Algorithms. Cambridge University Press. ISBN 978-0521432085.
[8] SOLAR NEBULA COLLAPSE TIMESCALE
Shu, F. H., Adams, F. C., and Lizano, S. (1987). Star formation in molecular clouds - Observation and theory. Annual Review of Astronomy and Astrophysics, 25, 23-81.
[9] GRAVITATIONAL FOCUSING (SAFRONOV)
Safronov, V. S. (1969). Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets. Nauka Press, Moscow. English translation: NASA TTF-677 (1972). The Safronov number theta sets the gravitational focusing factor: f_grav = 1 + theta, where theta = v_esc^2 / (2 * v_rel^2). The f_grav = 8 used here is representative of low-eccentricity disk orbits.
Simulation Architecture Note: This module implements a direct O(N^2) N-body gravitational integrator with velocity Verlet time-stepping. Internal units: length = 1 AU = 1.496e11 m; mass = 1 M_sun = 1.989e30 kg; time = 1 yr = 3.156e7 s, giving G = 4*pi^2 exactly. Physical (collision) radii are computed from bulk density: 3000 kg/m^3 for rocky bodies (inside snowline) and 1000 kg/m^3 for icy bodies (beyond snowline), consistent with Eq. 5. Display radii are scaled visually by cube-root of mass relative to a 3e-6 M_sun (Earth-mass) reference body shown at 1.2px, with a minimum of 0.8px and maximum of 6px. Individual super-planetesimals start near Moon-mass (1e-7 M_sun) and grow through inelastic mergers. Bodies are reclassified as Rocky Planet at 5e-7 M_sun (~0.17 Earth masses, inside snowline only), Proto-Planet at 5e-6 M_sun (~1.5 Earth masses), and Gas Giant at 3e-4 M_sun (~100 Earth masses). Collision detection uses a gravitational-focusing-enhanced radius threshold f_grav = 8 on the physical AU radii (Eq. 5). The simulation enters STABLE SYSTEM phase when fewer than 4 planetesimals remain and at least one proto-planet exists. In that phase each surviving world is annotated with its orbital radius (AU) and mass in Earth masses (Me).