Why Does the Sun Rotate So Slowly? The Mystery of the Missing Angular Momentum

What is Angular Momentum?

Angular Momentum is a vector quantity that measures the amount of rotation of a body relative to a point: \( \vec{L} = \vec{r} \times \vec{p} \)

\( \vec{L} \): Angular momentum, \( \vec{r} \): Position vector, which goes from the reference point (origin) to the moving body, \( \vec{p} \): Momentum of the body.

Theorem of Angular Momentum

The law of conservation of angular momentum is a fundamental principle of classical and quantum mechanics. It states that in an isolated system (where no net external force exerts a moment), the total angular momentum is conserved.

Physical Examples

• A skater who brings their arms closer to their body speeds up their rotation (reduction of the moment of inertia 𝐼, thus increase in angular velocity 𝜔, since 𝐿=𝐼𝜔).
• Planets accelerate as they approach the Sun (Kepler's 2nd law: conservation of angular momentum).
• A gyroscope maintains its axis of rotation fixed due to the conservation of angular momentum. If you try to tilt it, it reacts with a precession movement, illustrating the stability of angular momentum in the absence of a net external torque.
• In an isolated atom without perturbation, the orbital angular momentum of the electron is conserved. In the presence of interactions (such as the emission or absorption of photons), the orbital angular momentum can change, but the total angular momentum of the system (electron + field) remains conserved.

Origin of the Solar System Angular Momentum Paradox

In a scenario of a collapsing rotating proto-solar cloud, we expect the angular momentum to be conserved. However, simulations show that in the absence of angular momentum transfer mechanisms, the Sun should rotate much faster in a few hours instead of 27 days.

The angular momentum of the Solar System is largely concentrated in the planets, particularly the gas giants like Jupiter and Saturn. This fact is counterintuitive: the Sun contains more than 99.8% of the mass of the Solar System, but only about 2% of its total angular momentum. In contrast, Jupiter and Saturn alone account for more than 90% of this angular momentum.

Physical Definition of Orbital Angular Momentum

The orbital angular momentum \( L \) of a body of mass \( m \), moving in a circular orbit of radius \( r \) with a velocity \( v \), is given by: \( L = m \cdot r \cdot v \)

For a Keplerian orbit, \( L \) can be expressed in terms of the mass \( M \) of the central star (here the Sun), by: \( L = m \cdot \sqrt{G M r} \) where \( G \) is the gravitational constant.

The angular momentum paradox of the Solar System is an astrophysical enigma related to the unexpected distribution of angular momentum between the Sun and the planets.

Quantification of the Paradox

Source: NASA Planetary Fact Sheet, Ward & Canup 2002

Proposed Explanations for Resolving This Paradox

• Magnetic Braking: The young Sun would have had an intense magnetic field, interacting with the protoplanetary disk. This coupling would have transferred angular momentum from the Sun to the disk, slowing its rotation.
• Transfer by the Primitive Solar Wind: The solar winds coupled with the nascent magnetic field of the Sun would have carried away angular momentum.
• Planetary Migration: The giant planets (like Jupiter) would have migrated outward, increasing their orbital angular momentum at the expense of the Sun's.
• Tidal Effects with the Gas Disk: The tidal forces between the Sun and the protoplanetary disk would have redistributed the angular momentum.

The Main Reason for the Paradox: Magnetic Braking of the Young Sun

The main reason explaining the angular momentum paradox of the Solar System is magnetic braking, associated with the coupling between the magnetic field of the young Sun and the protoplanetary disk.

• Initial collapse of the solar nebula: A gas cloud collapses under gravity, forming a rapidly rotating proto-Sun (conservation of initial angular momentum). According to calculations, the Sun should rotate much faster than it does today.
• Key role of the magnetic field: The young Sun had a powerful magnetic field, connecting its surface to the surrounding gas disk. This field acted as a "brake": the magnetic field lines transferred the angular momentum from the Sun to the outer disk (where the planets were forming).

The Sun lost most of its rotational angular momentum to the disk, and then to the planets. The planets, forming at great distances, inherited most of the total angular momentum.

Why is This Explanation Preferred?

• Observations of young stars (such as T Tauri stars) show that they slow down over time, in agreement with this mechanism.
• Theoretical models and numerical simulations better reproduce the slow rotation of the Sun by including this magnetic coupling.