Authors

Arbatova Varvara Petrovna, Karpova Yesenia Alekseevna, Dagdelen Zeynap Rejepovna, Byugdanyuk Anna Vasilievna, Lyupp Sofya Romanovna

Peoples’ Friendship University of Russia (RUDN)

Moscow, Ordzhonikidze St. 3

Introduction

The formation of planetary systems is a key process in the evolution of the Universe

The theory explains the origin of the Solar System and exoplanets

Objective: To study the theoretical foundations of the formation of a planetary system from a gas and dust cloud

Origin of the Universe

Expansion and cooling → particles → atoms → galaxies → stars

Schmidt’s Theory (1944)

1. A rotating gas and dust cloud contracts
2. Rotation speed increases (law of conservation of angular momentum)
3. The cloud flattens into a disk
4. Rings form on the periphery → condense into planets

Laws of Motion

Kepler’s Third Law:

$$v \sim \frac{1}{\sqrt{r}}$$

Velocity decreases with distance from the center

Used to set initial velocities of particles

Gravitational Interaction

Potential Energy:

$$U_i = -\sum_{j \neq i} \frac{\gamma m_i m_j}{r_{ij}}$$

Gravity is a long-range force

Requires accounting for all particle pairs

Complexity O(N²) limits the model size

Repulsion and Friction Forces

Upon approach $b < R_i + R_j$:

Repulsion: $F^r(b) = k\left(\left(\frac{a}{b}\right)^8 - 1\right)$

Friction: $F_f = \beta W_{\perp} F^r(b) \mathbf{n}$

Friction is perpendicular to the radius vector, directed against the motion

Particle Rotation

Moment of inertia: $I = \frac{2}{5} m R^2$

Equation of rotation: $I\varepsilon = R\sum\frac{b}{R_i+R_j}F^f$

Rotational energy: $E_{\text{rot}} = \frac{I\omega^2}{2}$

Particle Coalescence

Upon complete coalescence:

$m = m_i + m_j$

$R = \sqrt[3]{R_i^3 + R_j^3}$

$\mathbf{r} = \frac{m_i\mathbf{r}_i + m_j\mathbf{r}_j}{m_i + m_j}$

$\mathbf{v} = \frac{m_i\mathbf{v}_i + m_j\mathbf{v}_j}{m_i + m_j}$

Conserves the mass and momentum of the system

Key Mechanisms

Conclusion

The theoretical foundations of the formation of the Solar System have been reviewed:

• Cosmological prehistory (Big Bang, supernovae)
• Schmidt’s theory of protoplanetary disk formation
• Physical mechanisms: gravity, repulsion, friction, rotation, coalescence

The obtained foundations will be used for numerical modeling