Cosmology and Astrophysics

Mathematical Model of the Ptolemaic System

This section describes the geocentric system using vector algebra and analytic geometry. It is designed to be directly translatable into code (e.g., JavaScript/GLSL) for a 3D simulation.

1. Vector Superposition Principle

In the Ptolemaic system, the complex motion of a planet is constructed by summing rotating vectors. This approach allows us to analytically derive the position of a planet $P$ relative to the stationary Earth $E$ at the origin $(0,0,0)$.

The position vector $\vec{P}$ of a planet is the sum of three fundamental vectors:

\[ \vec{P} = \vec{E} + \vec{D} + \vec{e} \]

Where:

$\vec{E}$ (Eccentricity): A static offset vector representing the displacement of the Deferent’s center from Earth.
$\vec{D}$ (Deferent): A rotating vector representing the motion of the Epicycle’s center along the Deferent.
$\vec{e}$ (Epicycle): A rapidly rotating vector representing the planet’s motion around the Epicycle center.

2. 3D Vector Implementation

To visualize this in a 3D simulation (where the $XY$ plane is the ecliptic and $Z$ is the altitude), we define the components of each vector.

Let:

$e$: Eccentricity (distance Earth-Deferent Center)
$R$: Radius of the Deferent
$r$: Radius of the Epicycle
$M$: Mean Anomaly (angle of the Deferent vector)
$\gamma$: Epicyclic Anomaly (angle of the Epicycle vector)
$i$: Inclination of the Deferent plane relative to the ecliptic

A. The Eccentricity Vector ($\vec{E}$)

This vector is constant and lies along the line of apsides (X-axis). It shifts the entire system off-center.

\[ \vec{E} = \begin{bmatrix} e \\ 0 \\ 0 \end{bmatrix} \]

B. The Deferent Vector ($\vec{D}$)

This vector rotates with angle $M$. Because the orbit is tilted by inclination $i$, the vector has a vertical $Z$ component.

\[ \vec{D} = \begin{bmatrix} R \cos(M) \\ R \sin(M) \cos(i) \\ R \sin(M) \sin(i) \end{bmatrix} \]

C. The Epicycle Vector ($\vec{e}$)

This vector rotates with angle $\gamma$. For simplicity, we assume the epicycle lies in the same plane as the deferent (though historically complex, this approximation suffices for basic visualization).

\[ \vec{e} = \begin{bmatrix} r \cos(\gamma) \\ r \sin(\gamma) \cos(i) \\ r \sin(\gamma) \sin(i) \end{bmatrix} \]

3. Final Planet Position

By summing the components, we obtain the parametric equations for the planet’s coordinates $(x, y, z)$ as a function of time:

\[ x(t) = e + R \cos(M) + r \cos(\gamma) \]

\[ y(t) = \left[ R \sin(M) + r \sin(\gamma) \right] \cos(i) \]

\[ z(t) = \left[ R \sin(M) + r \sin(\gamma) \right] \sin(i) \]

4. Retrograde Motion Condition

The looping motion (retrograde) occurs when the projected velocity of the planet opposes the general rotation of the system. This happens when the planet is at the perigee of its epicycle (closest to Earth).

Analytically, a loop forms if the linear velocity on the epicycle ($v_e$) exceeds the linear velocity on the deferent ($v_d$):

\[ v_e > v_d \quad \Rightarrow \quad \omega_{epi} \cdot r > \omega_{def} \cdot R \]

In 3D space, this manifests as the planet physically reversing its $x$ and $y$ velocity components relative to the observer, creating the characteristic loop on the celestial sphere.

5. Deviation from the Ecliptic (Celestial Latitude)

A critical feature of this 3D model is that the planet is not confined to the ecliptic line.

In a 2D simplification, the planet would always lie on the Sun’s path ($z=0$). However, due to the inclination term $\sin(i)$ in the $z(t)$ equation, the planet possesses a vertical component called Celestial Latitude.

Above the Ecliptic: When $z(t) > 0$, the planet is observed “above” the Sun’s path (North).
Below the Ecliptic: When $z(t) < 0$, the planet is observed “below” the Sun’s path (South).
On the Ecliptic: The planet is strictly on the ecliptic only when it crosses the orbital nodes (where $z(t) = 0$).

Consequently, an observer on Earth sees the planet wandering slightly up and down relative to the background stars of the Zodiac, rather than following a perfect line.

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Mathematical Model of the Copernican System (Heliocentric)

The system of Nicolaus Copernicus introduces a fundamental change of reference frame. By shifting the center of the Universe (and the coordinate system origin) to the Sun, we drastically simplify the mathematical description of planetary motions, eliminating the need for large epicycles to explain retrograde motion.

1. Main Assumptions

In an analytic approach, this model relies on the following axioms:

Sun ($S$): Is the stationary center of the system $(0,0,0)$.
Orbits: Planets (including Earth) move in orbits around the Sun.
- Historical Note: Copernicus still used circles (and small epicycles for speed correction), but in this simplified didactic model (and the later Keplerian one), we assume single closed curves (circles or ellipses).
Earth ($E$): Is no longer a static reference point but one of the planets, possessing its own motion vector.

2. Vector Description (Complexity Reduction)

Unlike the Ptolemaic system, where a planet’s position was the sum of three vectors (Eccentricity + Deferent + Epicycle), in the heliocentric system, the position of a planet $\vec{P}$ relative to the Sun is described by only one vector.

\[ \vec{P}_{helio} = \vec{O} \]

Where $\vec{O}$ is the orbital vector.

3. 3D Implementation (Heliocentric Coordinates)

For each planet (Mercury, Venus, Earth, Mars, etc.), we define its orbital parameters relative to the Sun.

Let: * $R$: Orbital radius (distance from the Sun) * $\Omega$: Angular velocity (related to the orbital period $T$ by $\Omega = 2\pi/T$) * $i$: Inclination (tilt of the orbit relative to the ecliptic plane—Earth’s orbital plane) * $t$: Time

The parametric equations for the planet’s position $[x, y, z]$ in the sun-centered system are:

\[ x(t) = R \cdot \cos(\Omega \cdot t) \]

\[ y(t) = R \cdot \sin(\Omega \cdot t) \cdot \cos(i) \]

\[ z(t) = R \cdot \sin(\Omega \cdot t) \cdot \sin(i) \]

Compared to Ptolemy, these equations are trivial. There are no nested trigonometric functions (cosine within cosine).

4. Transformation to the Geocentric Frame (Earth as Observer)

While the physics is simple in the heliocentric frame, our observations are made from Earth, which is itself moving. To understand what an astronomer sees, we must perform a vector coordinate transformation.

We define two position vectors relative to the Sun (Origin): 1. Earth’s Vector ($\vec{E}$): Assumed to be on the fundamental plane ($i_E = 0, z=0$). 2. Planet’s Vector ($\vec{P}$): The target planet we are observing.

\[ \vec{E}_{helio} = \begin{bmatrix} R_E \cos(\Omega_E \cdot t) \\ R_E \sin(\Omega_E \cdot t) \\ 0 \end{bmatrix}, \quad \vec{P}_{helio} = \begin{bmatrix} R_P \cos(\Omega_P \cdot t) \\ R_P \sin(\Omega_P \cdot t) \cos(i) \\ R_P \sin(\Omega_P \cdot t) \sin(i) \end{bmatrix} \]

To find the position of the Planet as seen from Earth ($\vec{P}_{obs}$), we simply subtract Earth’s vector from the Planet’s vector:

\[ \vec{P}_{obs} = \vec{P}_{helio} - \vec{E}_{helio} \]

The Parametric Equations for the Observer:

This vector subtraction yields the coordinates $(x', y', z')$ relative to the observer:

\[ x'(t) = \underbrace{R_P \cos(\Omega_P t)}_{\text{Planet Motion}} - \underbrace{R_E \cos(\Omega_E t)}_{\text{Parallax (Earth Motion)}} \]

\[ y'(t) = \underbrace{R_P \sin(\Omega_P t) \cos(i)}_{\text{Planet Motion}} - \underbrace{R_E \sin(\Omega_E t)}_{\text{Parallax (Earth Motion)}} \]

\[ z'(t) = R_P \sin(\Omega_P t) \sin(i) \quad \text{(Unchanged, as Earth has } z=0 \text{)} \]

Physical Interpretation: This mathematical operation is crucial. The term $-\vec{E}_{helio}$ adds a rotational component to the planet’s apparent motion.

In the Ptolemaic system, this component was modeled artificially by the Epicycle.
In the Copernican system, this component appears naturally as a result of the moving observer.

When Earth ($R_E, \Omega_E$) “overtakes” an outer planet ($R_P, \Omega_P$), the relative velocity momentarily reverses direction in the $x'$ or $y'$ axis, creating the loop (retrograde motion) without any physical change in the planet’s actual orbit.

5. Mechanism of Retrograde Motion (Overtaking)

In the Copernican system, loops in the sky are not “real” planetary motions (like Ptolemy’s epicycles). They are optical illusions resulting from the difference in linear velocities.

Planets closer to the Sun move faster (Kepler’s 3rd Law, though intuitively understood by Copernicus).
When Earth overtakes an outer planet (e.g., Mars) or is overtaken by an inner planet (e.g., Venus), the line of sight changes angle such that the planet appears to move backward against the background stars.

Mathematically, this does not require extra formulas in the physical model—it emerges naturally from vector subtraction, which we will discuss in the next section regarding the change of reference frame.

6. Vector Summary

Feature	Ptolemaic System	Copernican System
Center $(0,0,0)$	Earth	Sun
Vector Structure	$\vec{P} = \vec{E}_{ecc} + \vec{D}_{def} + \vec{e}_{epi}$	$\vec{P} = \vec{O}_{orb}$
Source of Loops	Real motion on a small circle (Epicycle)	Parallax effect (Earth’s motion)
Computational Complexity	High (summing multiple motions)	Low (single rotational motion)

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Johannes Kepler: The Laws of Planetary Motion

While Copernicus correctly identified the Sun as the center of the system, his model still relied on perfect circles and uniform speeds, which failed to perfectly match observations (specifically, the position of Mars). Johannes Kepler, analyzing the precise data of Tycho Brahe, abandoned the ancient dogma of circular motion.

He formulated three fundamental laws that describe orbital mechanics with mathematical precision.

1. Kepler’s First Law: The Law of Ellipses

The orbit of a planet is an ellipse with the Sun at one of the two foci.

In the Ptolemaic and Copernican models, the distance between a planet and the center was constant ($R$). In Kepler’s model, this distance $r$ varies continuously.

[Image of Kepler’s first law diagram]

Ellipse Geometry: An ellipse is defined by two points (foci). The sum of distances from any point on the curve to the two foci is constant.
Sun’s Position: The Sun is not at the center of the ellipse, but at one focus. The other focus is empty.

Mathematical Formulation (Polar Coordinates): If we place the Sun at the origin $(0,0)$, the distance $r$ as a function of the angle $\theta$ (True Anomaly) is:

\[ r(\theta) = \frac{a(1 - e^2)}{1 + e \cos \theta} \]

Where: * $a$: Semi-major axis (average distance). * $e$: Eccentricity (shape parameter, $0 \le e < 1$). * If $e=0$, the orbit is a circle. * As $e \to 1$, the orbit becomes a highly elongated ellipse.

2. Kepler’s Second Law: The Law of Equal Areas

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

[Image of Kepler’s second law equal areas]

This law destroyed the concept of “uniform circular motion.” It implies that orbital velocity is not constant.

Perihelion (Closest approach): The planet moves fastest.
Aphelion (Farthest point): The planet moves slowest.

Mathematical Formulation: The rate of change of the area $A$ swept by the radius vector is constant:

\[ \frac{dA}{dt} = \text{const} \]

In terms of physics (which Newton would later prove), this is a direct consequence of the Conservation of Angular Momentum:

\[ L = m \cdot r \cdot v_{\perp} = \text{const} \]

As the distance $r$ decreases, the velocity $v$ must increase to conserve momentum.

3. Kepler’s Third Law: The Law of Harmonies

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

This law connects the motion of all planets in the system into a unified physical mechanism. It implies that planets further from the Sun travel slower and have longer paths.

Mathematical Formulation:

\[ T^2 \propto a^3 \]

Or, comparing two planets (1 and 2):

\[ \left( \frac{T_1}{T_2} \right)^2 = \left( \frac{a_1}{a_2} \right)^3 \]

Where: * $T$: Orbital Period (time to complete one full orbit). * $a$: Semi-major axis.

Summary of the Evolution of Models

Feature	Ptolemy	Copernicus	Kepler
Shape	Circle (+ Epicycles)	Circle (+ Epicycles)	Ellipse
Sun Position	Orbiting Earth	Center (Geometric)	Focus (Off-center)
Velocity	Uniform (relative to Equant)	Uniform	Variable (Max at Perihelion)
Mechanism	Geometry	Geometry	Physics (Precursor)

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--- title: Cosmology and Astrophysics format: html: theme: flatly grid: body-width: 1000px # Domyślnie jest to ok. 800-900px margin-width: 250px toc: true toc-depth: 3 highlight-style: tango code-line-numbers: true code-fold: true code-summary: "Show the code" code-tools: true code-block-bg: "rgba(42, 174, 42, 0.02)" code-block-border-left: "#2aae2a" code-language-label: true css: styles.css math: mathjax self-contained: true other-links: - text: Main page href: https://dchorazkiewicz.github.io/Mathematics_Physics_Lectures --- ## Mathematical Model of the Ptolemaic System This section describes the geocentric system using vector algebra and analytic geometry. It is designed to be directly translatable into code (e.g., JavaScript/GLSL) for a 3D simulation. ### 1. Vector Superposition Principle In the Ptolemaic system, the complex motion of a planet is constructed by summing rotating vectors. This approach allows us to analytically derive the position of a planet $P$ relative to the stationary Earth $E$ at the origin $(0,0,0)$. The position vector $\vec{P}$ of a planet is the sum of three fundamental vectors: $$ \vec{P} = \vec{E} + \vec{D} + \vec{e} $$ Where: * **$\vec{E}$ (Eccentricity):** A static offset vector representing the displacement of the Deferent's center from Earth. * **$\vec{D}$ (Deferent):** A rotating vector representing the motion of the Epicycle's center along the Deferent. * **$\vec{e}$ (Epicycle):** A rapidly rotating vector representing the planet's motion around the Epicycle center. ### 2. 3D Vector Implementation To visualize this in a 3D simulation (where the $XY$ plane is the ecliptic and $Z$ is the altitude), we define the components of each vector. Let: * $e$: Eccentricity (distance Earth-Deferent Center) * $R$: Radius of the Deferent * $r$: Radius of the Epicycle * $M$: Mean Anomaly (angle of the Deferent vector) * $\gamma$: Epicyclic Anomaly (angle of the Epicycle vector) * $i$: Inclination of the Deferent plane relative to the ecliptic #### A. The Eccentricity Vector ($\vec{E}$) This vector is constant and lies along the line of apsides (X-axis). It shifts the entire system off-center. $$ \vec{E} = \begin{bmatrix} e \\ 0 \\ 0 \end{bmatrix} $$ #### B. The Deferent Vector ($\vec{D}$) This vector rotates with angle $M$. Because the orbit is tilted by inclination $i$, the vector has a vertical $Z$ component. $$ \vec{D} = \begin{bmatrix} R \cos(M) \\ R \sin(M) \cos(i) \\ R \sin(M) \sin(i) \end{bmatrix} $$ #### C. The Epicycle Vector ($\vec{e}$) This vector rotates with angle $\gamma$. For simplicity, we assume the epicycle lies in the same plane as the deferent (though historically complex, this approximation suffices for basic visualization). $$ \vec{e} = \begin{bmatrix} r \cos(\gamma) \\ r \sin(\gamma) \cos(i) \\ r \sin(\gamma) \sin(i) \end{bmatrix} $$ ### 3. Final Planet Position By summing the components, we obtain the parametric equations for the planet's coordinates $(x, y, z)$ as a function of time: $$ x(t) = e + R \cos(M) + r \cos(\gamma) $$ $$ y(t) = \left[ R \sin(M) + r \sin(\gamma) \right] \cos(i) $$ $$ z(t) = \left[ R \sin(M) + r \sin(\gamma) \right] \sin(i) $$ ### 4. Retrograde Motion Condition The looping motion (retrograde) occurs when the projected velocity of the planet opposes the general rotation of the system. This happens when the planet is at the perigee of its epicycle (closest to Earth). Analytically, a loop forms if the linear velocity on the epicycle ($v_e$) exceeds the linear velocity on the deferent ($v_d$): $$ v_e > v_d \quad \Rightarrow \quad \omega_{epi} \cdot r > \omega_{def} \cdot R $$ In 3D space, this manifests as the planet physically reversing its $x$ and $y$ velocity components relative to the observer, creating the characteristic loop on the celestial sphere. ### 5. Deviation from the Ecliptic (Celestial Latitude) A critical feature of this 3D model is that the planet is **not confined to the ecliptic line**. In a 2D simplification, the planet would always lie on the Sun's path ($z=0$). However, due to the inclination term $\sin(i)$ in the $z(t)$ equation, the planet possesses a vertical component called **Celestial Latitude**. * **Above the Ecliptic:** When $z(t) > 0$, the planet is observed "above" the Sun's path (North). * **Below the Ecliptic:** When $z(t) < 0$, the planet is observed "below" the Sun's path (South). * **On the Ecliptic:** The planet is strictly on the ecliptic **only** when it crosses the orbital nodes (where $z(t) = 0$). Consequently, an observer on Earth sees the planet wandering slightly up and down relative to the background stars of the Zodiac, rather than following a perfect line. <iframe src="html_anim/cosmology/ptolomeo.html" width="100%" height="700px"></iframe> [html_anim/cosmology/ptolomeo.html](html_anim/cosmology/ptolomeo.html) ## Mathematical Model of the Copernican System (Heliocentric) The system of Nicolaus Copernicus introduces a fundamental change of reference frame. By shifting the center of the Universe (and the coordinate system origin) to the Sun, we drastically simplify the mathematical description of planetary motions, eliminating the need for large epicycles to explain retrograde motion. ### 1. Main Assumptions In an analytic approach, this model relies on the following axioms: * **Sun ($S$):** Is the stationary center of the system $(0,0,0)$. * **Orbits:** Planets (including Earth) move in orbits around the Sun. * *Historical Note:* Copernicus still used circles (and small epicycles for speed correction), but in this simplified didactic model (and the later Keplerian one), we assume single closed curves (circles or ellipses). * **Earth ($E$):** Is no longer a static reference point but one of the planets, possessing its own motion vector. ### 2. Vector Description (Complexity Reduction) Unlike the Ptolemaic system, where a planet's position was the sum of three vectors (Eccentricity + Deferent + Epicycle), in the heliocentric system, the position of a planet $\vec{P}$ relative to the Sun is described by **only one vector**. $$ \vec{P}_{helio} = \vec{O} $$ Where $\vec{O}$ is the orbital vector. ### 3. 3D Implementation (Heliocentric Coordinates) For each planet (Mercury, Venus, Earth, Mars, etc.), we define its orbital parameters relative to the Sun. Let: * $R$: Orbital radius (distance from the Sun) * $\Omega$: Angular velocity (related to the orbital period $T$ by $\Omega = 2\pi/T$) * $i$: Inclination (tilt of the orbit relative to the ecliptic plane—Earth's orbital plane) * $t$: Time The parametric equations for the planet's position $[x, y, z]$ in the sun-centered system are: $$ x(t) = R \cdot \cos(\Omega \cdot t) $$ $$ y(t) = R \cdot \sin(\Omega \cdot t) \cdot \cos(i) $$ $$ z(t) = R \cdot \sin(\Omega \cdot t) \cdot \sin(i) $$ *Compared to Ptolemy, these equations are trivial. There are no nested trigonometric functions (cosine within cosine).* ### 4. Transformation to the Geocentric Frame (Earth as Observer) While the physics is simple in the heliocentric frame, our observations are made from Earth, which is itself moving. To understand what an astronomer sees, we must perform a **vector coordinate transformation**. We define two position vectors relative to the Sun (Origin): 1. **Earth's Vector ($\vec{E}$):** Assumed to be on the fundamental plane ($i_E = 0, z=0$). 2. **Planet's Vector ($\vec{P}$):** The target planet we are observing. $$ \vec{E}_{helio} = \begin{bmatrix} R_E \cos(\Omega_E \cdot t) \\ R_E \sin(\Omega_E \cdot t) \\ 0 \end{bmatrix}, \quad \vec{P}_{helio} = \begin{bmatrix} R_P \cos(\Omega_P \cdot t) \\ R_P \sin(\Omega_P \cdot t) \cos(i) \\ R_P \sin(\Omega_P \cdot t) \sin(i) \end{bmatrix} $$ To find the position of the Planet as seen from Earth ($\vec{P}_{obs}$), we simply subtract Earth's vector from the Planet's vector: $$ \vec{P}_{obs} = \vec{P}_{helio} - \vec{E}_{helio} $$ **The Parametric Equations for the Observer:** This vector subtraction yields the coordinates $(x', y', z')$ relative to the observer: $$ x'(t) = \underbrace{R_P \cos(\Omega_P t)}_{\text{Planet Motion}} - \underbrace{R_E \cos(\Omega_E t)}_{\text{Parallax (Earth Motion)}} $$ $$ y'(t) = \underbrace{R_P \sin(\Omega_P t) \cos(i)}_{\text{Planet Motion}} - \underbrace{R_E \sin(\Omega_E t)}_{\text{Parallax (Earth Motion)}} $$ $$ z'(t) = R_P \sin(\Omega_P t) \sin(i) \quad \text{(Unchanged, as Earth has } z=0 \text{)} $$ **Physical Interpretation:** This mathematical operation is crucial. The term $-\vec{E}_{helio}$ adds a rotational component to the planet's apparent motion. * In the Ptolemaic system, this component was modeled artificially by the **Epicycle**. * In the Copernican system, this component appears naturally as a result of the **moving observer**. When Earth ($R_E, \Omega_E$) "overtakes" an outer planet ($R_P, \Omega_P$), the relative velocity momentarily reverses direction in the $x'$ or $y'$ axis, creating the loop (retrograde motion) without any physical change in the planet's actual orbit. ### 5. Mechanism of Retrograde Motion (Overtaking) In the Copernican system, loops in the sky are not "real" planetary motions (like Ptolemy's epicycles). They are optical illusions resulting from the difference in linear velocities. * Planets closer to the Sun move faster (Kepler's 3rd Law, though intuitively understood by Copernicus). * When Earth overtakes an outer planet (e.g., Mars) or is overtaken by an inner planet (e.g., Venus), the line of sight changes angle such that the planet appears to move backward against the background stars. Mathematically, this does not require extra formulas in the physical model—it emerges naturally from vector subtraction, which we will discuss in the next section regarding the change of reference frame. ### 6. Vector Summary | Feature | Ptolemaic System | Copernican System | | :--- | :--- | :--- | | **Center $(0,0,0)$** | Earth | Sun | | **Vector Structure** | $\vec{P} = \vec{E}_{ecc} + \vec{D}_{def} + \vec{e}_{epi}$ | $\vec{P} = \vec{O}_{orb}$ | | **Source of Loops** | Real motion on a small circle (Epicycle) | Parallax effect (Earth's motion) | | **Computational Complexity** | High (summing multiple motions) | Low (single rotational motion) | <iframe src="html_anim/cosmology/copernicus.html" width="100%" height="700px"></iframe> [html_anim/cosmology/copernicus.html](html_anim/cosmology/copernicus.html) <iframe src="html_anim/cosmology/copernicus_from_earth.html" width="100%" height="700px"></iframe> [html_anim/cosmology/copernicus_from_earth.html](html_anim/cosmology/copernicus_from_earth.html) ## Johannes Kepler: The Laws of Planetary Motion While Copernicus correctly identified the Sun as the center of the system, his model still relied on perfect circles and uniform speeds, which failed to perfectly match observations (specifically, the position of Mars). Johannes Kepler, analyzing the precise data of Tycho Brahe, abandoned the ancient dogma of circular motion. He formulated three fundamental laws that describe orbital mechanics with mathematical precision. ### 1. Kepler's First Law: The Law of Ellipses **The orbit of a planet is an ellipse with the Sun at one of the two foci.** In the Ptolemaic and Copernican models, the distance between a planet and the center was constant ($R$). In Kepler's model, this distance $r$ varies continuously. [Image of Kepler's first law diagram] * **Ellipse Geometry:** An ellipse is defined by two points (foci). The sum of distances from any point on the curve to the two foci is constant. * **Sun's Position:** The Sun is **not** at the center of the ellipse, but at one focus. The other focus is empty. **Mathematical Formulation (Polar Coordinates):** If we place the Sun at the origin $(0,0)$, the distance $r$ as a function of the angle $\theta$ (True Anomaly) is: $$ r(\theta) = \frac{a(1 - e^2)}{1 + e \cos \theta} $$ Where: * $a$: **Semi-major axis** (average distance). * $e$: **Eccentricity** (shape parameter, $0 \le e < 1$). * If $e=0$, the orbit is a circle. * As $e \to 1$, the orbit becomes a highly elongated ellipse. ### 2. Kepler's Second Law: The Law of Equal Areas **A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.** [Image of Kepler's second law equal areas] This law destroyed the concept of "uniform circular motion." It implies that **orbital velocity is not constant**. * **Perihelion (Closest approach):** The planet moves fastest. * **Aphelion (Farthest point):** The planet moves slowest. **Mathematical Formulation:** The rate of change of the area $A$ swept by the radius vector is constant: $$ \frac{dA}{dt} = \text{const} $$ In terms of physics (which Newton would later prove), this is a direct consequence of the **Conservation of Angular Momentum**: $$ L = m \cdot r \cdot v_{\perp} = \text{const} $$ As the distance $r$ decreases, the velocity $v$ must increase to conserve momentum. ### 3. Kepler's Third Law: The Law of Harmonies **The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.** This law connects the motion of all planets in the system into a unified physical mechanism. It implies that planets further from the Sun travel slower and have longer paths. **Mathematical Formulation:** $$ T^2 \propto a^3 $$ Or, comparing two planets (1 and 2): $$ \left( \frac{T_1}{T_2} \right)^2 = \left( \frac{a_1}{a_2} \right)^3 $$ Where: * $T$: Orbital Period (time to complete one full orbit). * $a$: Semi-major axis. ### Summary of the Evolution of Models | Feature | Ptolemy | Copernicus | Kepler | | :--- | :--- | :--- | :--- | | **Shape** | Circle (+ Epicycles) | Circle (+ Epicycles) | **Ellipse** | | **Sun Position** | Orbiting Earth | Center (Geometric) | **Focus (Off-center)** | | **Velocity** | Uniform (relative to Equant) | Uniform | **Variable** (Max at Perihelion) | | **Mechanism** | Geometry | Geometry | **Physics (Precursor)** | <iframe src="html_anim/cosmology/kepler.html" width="100%" height="700px"></iframe> [html_anim/cosmology/kepler.html](html_anim/cosmology/kepler.html)