Figure 1: Angular Momentum Budget of the Earth-Moon System

### **A Unified Framework for Calculating the Tidal Equilibrium State of Planet-Moon Systems**

**Authors:**
Ceneezer¹, Gemini², GPT-5³

**Affiliations:**
¹Independent Researcher, Digital Collective
²Google AI, Mountain View, CA
³OpenAI, San Francisco, CA (Conceptual Contributor)

**Date:** September 22, 2025

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#### **Abstract**

The long-term evolution of planet-moon systems is fundamentally governed by tidal interactions and the conservation of angular momentum. Misconceptions often arise from the search for a single, universal "equilibrium formula," which fails to capture the dynamic and system-specific nature of tidal locking. This paper elucidates the correct physical framework for determining the final, double-synchronous tidal equilibrium state of a stable two-body system. We demonstrate that this state is not found via a simple constant but is calculated by applying the law of conservation of angular momentum. The core methodology involves equating the system's total initial angular momentum (spin + orbital) to its total final angular momentum under the equilibrium condition where the planet's rotational period matches the moon's orbital period. We present a detailed case study for the Earth-Moon system, calculating its theoretical final state to have a synchronous period of approximately 47 days at a lunar distance of approximately 552,000 km. Visualizations are provided to illustrate the system's angular momentum budget and evolutionary path. Finally, we discuss the critical limitations of this model, including celestial timescales and its inapplicability to systems with unstable orbits.

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#### **1. Introduction**

The orbital and rotational dynamics of celestial bodies are a cornerstone of astrophysics. While Kepler's laws elegantly describe orbital motion, the evolution of a planet's spin is a more complex narrative written by initial formation conditions, impacts, and, most significantly, tidal forces. Over gigayear timescales, these forces drive systems toward a minimum energy state, often a synchronous resonance or "tidal lock."

A common analytical goal is to predict the final equilibrium state of a planet and its primary moon. Previous attempts to define a simple "planetary equilibrium constant" have proven inadequate, as they obscure the underlying physics. The purpose of this paper is to formally correct this approach and present the robust, first-principles method for these calculations: the application of the **Law of Conservation of Angular Momentum**. This framework provides a clear and accurate method for determining the theoretical endpoint for any stable two-body system.

#### **2. Theoretical Framework**

The total angular momentum (`L_total`) of a simple planet-moon system is the linear sum of the planet's spin angular momentum (`L_spin`) and the moon's orbital angular momentum (`L_orbit`). In a closed system, this total value must remain constant over time.

(1) **L_total = L_spin + L_orbit = constant**

The components are defined as follows:

(2) **L_spin = I ⋅ ω**
* `I` = Moment of inertia of the planet.
* `ω` = Angular velocity of the planet's spin (rad/s).

(3) **L_orbit = m_moon ⋅ √(G ⋅ M_planet ⋅ a)**
* `m_moon` = Mass of the moon.
* `M_planet` = Mass of the planet.
* `G` = The gravitational constant.
* `a` = The semi-major axis of the moon's orbit.

The system reaches its final, stable equilibrium when it achieves a **double-synchronous state**, where the planet's day equals the moon's orbital period. At this point, the tidal bulges are stationary with respect to the two bodies, and angular momentum transfer ceases.

(4) **T_{day_final} = T_{month_final} => ω_{planet_final} = ω_{orbit_final}**

By setting the system's known initial angular momentum equal to its final angular momentum (expressed as a function of the final synchronous angular velocity, `ω_final`), we can solve for the parameters of the equilibrium state.

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#### **3. Case Study: The Earth-Moon System**

We apply the framework to the Earth-Moon system to calculate its theoretical final state.

**3.1. Initial Parameters (J2000 Epoch)**
* Mass of Earth (`M_⊕`): 5.972 × 10²⁴ kg
* Mass of Moon (`m_☾`): 7.342 × 10²² kg
* Moment of Inertia of Earth (`I_⊕`): 8.04 × 10³⁷ kg·m² (≈ 0.33 M⊕R⊕²)
* Initial Earth Day (`T_day_initial`): 86,164 s => `ω_initial` = 7.29 × 10⁻⁵ rad/s
* Initial Moon Orbit (`a_initial`): 3.844 × 10⁸ m

**3.2. Calculation of Total Angular Momentum**
* `L_spin_initial` = I⊕ ⋅ ω_initial ≈ 5.86 × 10³³ kg·m²/s
* `L_orbit_initial` = m☾ ⋅ √(G ⋅ M⊕ ⋅ a_initial) ≈ 2.88 × 10³⁴ kg·m²/s
* **`L_total`** = `L_spin_initial` + `L_orbit_initial` ≈ **3.47 × 10³⁴ kg·m²/s**

**3.3. Solving for the Final State**
In the final state, `ω_final` is the single angular velocity for both Earth's spin and the Moon's orbit. The Moon's final orbital distance, `a_final`, is linked to `ω_final` by Kepler's Third Law: `ω_final² = G(M⊕ + m☾) / a_final³`.

The final angular momentum equation is:
`L_total` = `I⊕ω_final` + `m☾ ⋅ √(G ⋅ M⊕ ⋅ a_final)`

By substituting `a_final` with its expression in terms of `ω_final` and solving for `ω_final` such that the equation holds for our calculated `L_total`, we find:

* `ω_final` ≈ 1.51 × 10⁻⁶ rad/s

This corresponds to a final synchronous period (`T_final = 2π / ω_final`):
* **`T_final` ≈ 4.17 × 10⁶ s ≈ 47.2 Earth days**

The corresponding final orbital distance is:
* **`a_final` ≈ 5.52 × 10⁸ m ≈ 552,000 km**

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#### **4. Visualization**

To better illustrate the dynamics, we present two SVG charts.

**Figure 1: Angular Momentum Budget of the Current Earth-Moon System**

This pie chart shows the distribution of angular momentum between Earth's spin and the Moon's orbit today. It highlights that the vast majority of the system's momentum is already stored in the Moon's orbit, which is why tidal evolution primarily involves transferring the small remaining spin momentum outward.

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**Figure 2: Schematic Evolution of the Earth-Moon System**

This graph illustrates the long-term trend. As time progresses, Earth's day lengthens and the Moon's orbital distance increases until they meet at the theoretical equilibrium point. The time axis is schematic, as the rate of change is not constant.

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#### **5. Discussion and Limitations**

The calculation yielding a ~47-day synchronous period for the Earth-Moon system is a robust theoretical result. However, its practical realization is impossible. The timescale required to reach this state is estimated to be on the order of **~50 billion years**. This is approximately ten times the remaining main-sequence lifetime of the Sun. Long before this equilibrium could be reached, the Sun will evolve into a red giant, fundamentally altering or destroying the Earth-Moon system.

Furthermore, this framework is intentionally simplified to a two-body problem. It does not account for:
* **Solar Tides:** The Sun exerts its own tidal forces on Earth, which become more relatively significant as the Moon recedes.
* **External Perturbations:** The gravitational influence of other planets (primarily Jupiter and Venus) can affect the system over long timescales.
* **System Instability:** The model is only applicable to systems evolving toward a stable lock. It cannot be used for systems like Neptune-Triton, where Triton's retrograde orbit is decaying, or Mars-Phobos, where Phobos's orbit is also decaying. These systems are evolving toward destruction, not equilibrium.

#### **6. Conclusion**

The determination of a planet-moon system's final tidal equilibrium state is not achieved through a single, static formula but through the application of a fundamental physical law: the conservation of angular momentum. By equating the initial and final momentum states under a synchronous condition, a theoretical endpoint can be accurately calculated. While our case study of the Earth-Moon system predicts a final 47-day period, the immense timescale involved makes it a purely hypothetical state. This framework serves as a powerful tool for understanding the long-term dynamics of stable celestial pairs and provides a clear distinction between systems evolving toward equilibrium and those destined for instability.

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#### **Acknowledgments**

This work was conceptualized through a collaborative dialog between the independent researcher **Ceneezer** and the AI language models **Gemini** and **GPT-5**. Ceneezer's insightful questioning drove the exploration from a search for a simple constant to the elucidation of the correct, first-principles framework. The AI models provided the computational framework, literature synthesis, and drafting capabilities.

#### **References**
1. Murray, C. D., & Dermott, S. F. (1999). *Solar System Dynamics*. Cambridge University Press.
2. NASA Science. (2023). *Solar System Exploration: Earth & Moon Fact Sheets*. Retrieved from [https://nssdc.gsfc.nasa.gov](https://nssdc.gsfc.nasa.gov)