Chapter 34: Deriving Stellar Motion from First Principles

Lia gazed at the parchment on the table, densely covered with formulas and deductions.

Its title read: ‘Principles of the Unification of Celestial and Terrestrial Motion’.

The title was grand, and its content revolutionary.

With a single formula, it forged a connection between the stars in the heavens and an apple falling to earth.

Yet, Lia knew this was still insufficient.

A portion of the paper’s foundation rested upon an empirical law—one named ‘The Laws of Stellar Motion’, akin to Kepler’s laws on Earth.

To erect a truly unshakeable theoretical edifice, this empirical foundation had to be replaced with an impregnable bedrock of pure logic.

She intended to derive the laws of stellar motion anew, from first principles, using her three fundamental laws and the principle of universal gravitation.

Lia picked up her pen, adding a new chapter title to the end of the treatise.

‘Chapter Four: Derivation of the Laws of Stellar Motion from First Principles’

“The ‘Laws of Stellar Motion’ discussed previously are empirical observations summarized by scholars through prolonged study.

While they describe phenomena, they fail to explain their underlying causes.

The following content will demonstrate that these laws are not isolated principles, but rather the inevitable consequence of my proposed ‘Three Laws of Motion’ and the ‘Law of Universal Gravitation’.”

“I. The Inevitability of Orbital Shape—Why Elliptical?”

“We know that planets orbit the sun because the sun exerts a gravitational force upon them.

According to the Second Law of Motion, this force dictates the planet’s acceleration.

We can thus formulate the equation of motion:”

F_grav = m_planet × a

“Substituting the universal gravitation formula and employing the language of calculus to describe acceleration (where acceleration is the second derivative of position with respect to time), we arrive at a vector differential equation describing planetary motion.”

G × (M_sun × m_planet) / r² = m_planet × (d²r/dt²)

Lia rapidly performed calculations on a scrap piece of paper.

The process of solving this equation was considerably intricate, involving complex techniques in vector calculus, which necessitated Lia dedicating a substantial portion of her explanation to it.

“The mathematical solution to this equation demonstrates that, under the influence of a central force inversely proportional to the square of the distance, an object’s trajectory must invariably be a type of ‘conic section’—specifically, a circle, ellipse, parabola, or hyperbola.”

“From astronomical observations, we understand that planetary orbits are both closed and periodic.

Among all conic sections, only circles and ellipses are closed.

A circle, moreover, is merely a special case of an ellipse where its two foci coincide.”

“Thus, theory dictates: the orbit of a planet must necessarily be elliptical, and the central celestial body providing the gravitational force (the sun) must invariably reside at one of that ellipse’s foci.”

Having penned this sentence, Lia paused her writing.

The ‘First Law of Stellar Motion’ was no longer a conjecture derived from observation, but a certainty deduced from first principles.

Next came the Second Law, the one concerning area velocity.

“II. Conservation of Area Velocity—A Manifestation of Angular Momentum Conservation”

“To analyze orbital motion in greater depth, I must introduce a new physical quantity: angular momentum.”

“Angular Momentum (symbol L): A physical quantity describing an object’s rotational state of motion.

It is a vector whose magnitude equals the product of the object’s mass, its distance from the center of rotation, and the component of its velocity perpendicular to that distance.

Its direction is perpendicular to the plane of the object’s orbit.”

“It can be concisely expressed using the vector cross product as: L = r × p, where r is the position vector and p is the momentum (p = m × v).”

With the definition complete, the derivation began.

“Now, let us consider the rate of change of angular momentum with respect to time, which is dL/dt.”

“According to the rules of calculus for differentiation, dL/dt = (dr/dt × p) + (r × dp/dt).”

“The first term, (dr/dt × p), which is (v × m·v), is invariably zero because the velocity vector v is parallel to itself, making their cross product zero.”

“The second term, (r × dp/dt), by the Second Law of Motion, means that the time rate of change of momentum, dp/dt, is the net external force F acting on the object.

Therefore, (r × dp/dt) = r × F.

This quantity we term ‘torque’.”

“In the case of universal gravitation, the direction of the force always lies along the line connecting the sun and the planet (i.e., in the direction of the position vector r).

This type of force is known as a ‘central force’.

When the force F is collinear with the position vector r, their cross product (torque) is also invariably zero.”

“In summation, we arrive at a crucial conclusion: dL/dt = 0.”

“This signifies that for a planet moving within the sun’s gravitational field, its angular momentum remains a constant quantity.

I term this the ‘Law of Conservation of Angular Momentum’.”

“What, then, is the relationship between this and the area swept by a planet in a unit of time?”

Lia sketched a simple diagram alongside: a planet moving from point A to point B within an infinitesimally short time dt, forming a slender triangle with the sun S.

“It can be proven through calculus that the area swept by the line connecting the planet and the sun in a unit of time (area velocity dS/dt) is equal in magnitude to the planet’s angular momentum L divided by twice its mass (2m).”

dS/dt = |L| / (2m)

“Since both angular momentum L and mass m are constant, the area velocity dS/dt must also be a constant.”

“This theoretically proves the ‘Second Law of Stellar Motion’: the line connecting a planet to the sun sweeps out equal areas in equal times.

Its essence is a direct manifestation of the conservation of angular momentum within a central force field.”

Lia felt her thoughts clearer than they had ever been.

Finally, and most critically, came the law of periods.

“III. The Relationship Between Period and Orbit—A Refinement and Elevation of Empirical Law”

“The existing ‘Third Law’ (T²∝r³) is an empirical formula based on approximately circular orbits.

My theory, however, will provide a more precise and universally applicable description.”

She utilized the conclusions from the preceding two sections as her tools.

“We know that the time it takes for a planet to complete one revolution (period T) is equal to its total orbital area S, divided by its constant area velocity dS/dt.”

T = S / (dS/dt)

“For an elliptical orbit, its area is S = πab (where ‘a’ is the semi-major axis and ‘b’ is the semi-minor axis).”

“And the area velocity dS/dt = L / (2m).”

“Substituting these, we obtain T = 2mπab / L.”

“The subsequent derivation is rather complex, requiring the integration of the law of conservation of orbital energy to ultimately establish the relationship between angular momentum L, the semi-major axis ‘a’, and the semi-minor axis ‘b’.”

She took up her pen and meticulously wrote down the rigorous derivation, arriving at the formula that would redefine the astronomy of the entire world.

T² = (4π² / (G × M_sun)) × a³

“This formula demonstrates: the square of a planet’s orbital period is directly proportional to the cube of its orbit’s ‘semi-major axis’.”

Lia intentionally underlined the words ‘semi-major axis’.

“This rectifies the previous vague understanding that used orbital radius as the standard of measurement.

More significantly, the formula explicitly details the composition of the proportionality constant: it is no longer an empirical constant ‘k’ requiring repeated measurements, but a theoretical value uniquely determined by the universal gravitational constant G and the mass of the central celestial body (the sun), M_sun.”

“This implies that, provided we can precisely determine the value of G through experimentation, we can then, by observing a planet’s orbital period and semi-major axis, inversely calculate the mass of the sun!”

“Conversely, if the sun’s mass is known, we can precisely predict the orbital period of any unknown planet simply by knowing the semi-major axis of its orbit.”