Unit Conversions, Newton's Laws, and Orbital Dynamics

Non-Standard Units and Unit Invariance

• Arbitrary and Non-Standard Units of Measurement:

  • Length and distance can be expressed using non-standard or arbitrary units outside of standard metric or imperial systems (such as feet, inches, centimeters, or meters).

  • Human Head Heights ("Heads"): As a general rule of thumb, human body height is often approximated as being seven heads tall. However, individual head sizes scale with overall body dimensions (a smaller individual possesses a smaller head, whereas a larger individual possesses a larger head). Empirical measurement yields individual variations, such as a height equivalent to $5.7$\text{ heads}.

  • Conversion via Intermediate Units: To calculate personal height in an arbitrary unit, measure the arbitrary unit using an intermediate standard (such as tickets), measure total height in that same intermediate standard, and compute the mathematical ratio.

• Pedagogical Purpose of Conversions:

  • Unit conversion exercises emphasize understanding relational scale and conversion factors rather than memorization.

  • Query Guidelines: Look up raw fundamental conversion factors (e.g., determining the number of centimeters in a furlong or inches in a light-year) and complete the algebraic steps manually, rather than relying on automated direct conversions (e.g., directly querying the total conversion of $66$\text{ inches} into nautical miles).

• Empirical Height Conversions:

  • $9$\text{ water bottles tall}.

  • $0.0091$\text{ furlongs tall}.

  • $0.0011$\text{ miles tall}.

  • $0.52$\text{ albatross wingspans tall}.

  • Approximately $1$\text{ elephant trunk tall} (accounting for standard rounding tolerance, as an elephant trunk is roughly equal to human height).

• Principle of Unit Invariance:

  • A physical quantity, such as height, remains constant regardless of the unit system selected to express it. This property is defined as unit invariance.

  • Conversion factors are mathematically equal to $1$ (e.g., $\frac{1\,\text{albatross wingspan}}{\text{equivalent centimeters}}$). Multiplying a quantity by a unit conversion factor changes the units of representation without altering the underlying physical magnitude.

• Contextual Selection of Units:

  • In astrophysics and modeling, different units are selected based on convenience:

    • Interstellar distances and stellar explosion shockwave models may be calculated in centimeters ($\text{cm}$), meters ($\text{m}$), or astronomical units ($\text{AU}$).

    • To evaluate inter-stellar distances relative to the Earth-Sun scale, distances computed in centimeters ($\text{cm}$) are converted into astronomical units ($\text{AU}$).

Kepler's Laws and Newton's Generalization

• Kepler's Third Law:

  • Expressed mathematically as:         $P^2 = a^3$

  • $P$ = Orbital period measured in years ($\text{years}$).

  • $a$ = Semi-major axis measured in astronomical units ($\text{AU}$).

  • When period ($P$) is in years ($\text{years}$) and semi-major axis ($a$) is in astronomical units ($\text{AU}$), the constant of proportionality $k$ has a numerical magnitude of $1$, simplifying orbital calculations.

• Newton's Generalization of Kepler's Third Law:

  • Isaac Newton incorporated mass into Kepler's framework, expanding its application from solar planets to any two gravitationally bound bodies in mutual orbit:         $(M + m) \times P^2 = a^3$

  • $M$ = Mass of the primary or central object.

  • $m$ = Mass of the secondary or orbiting object.

  • $P$ = Orbital period in years ($\text{years}$).

  • $a$ = Semi-major axis in astronomical units ($\text{AU}$).

Solar Mass Units and Planetary Systems

• Definition of Solar Mass Units:

  • To maintain the mathematical simplicity where $(M + m) \times P^2 = a^3$ holds without extra proportionality constants, mass is defined in solar mass units ($\text{M}_{\odot}$).

  • The mass of the Sun ($M$) is set to exactly $1\,\text{solar mass unit}$ ($1\,\text{M}_{\odot}$).

• Mass Evaluation in the Solar System:

  • For the Earth orbiting the Sun:

    • Central solar mass $M = 1\,\text{M}_{\odot}$.

    • Secondary Earth mass $m = 3.00274 \times 10^{-6}\,\text{M}_{\odot}$ (or $0.00000300274\,\text{M}_{\odot}$).

    • Sum of masses: $(M + m) = 1 + 0.00000300274 = 1.00000300274\,\text{M}_{\odot}$.

  • Because $m_{\text{Earth}}$ is negligible compared to $1\,\text{M}_{\odot}$, it functions as a rounding error and is dropped in general calculations ($M + m \approx 1$).

  • For comets (e.g., Halley's Comet), $m$ is even smaller than Earth's mass, making $m \approx 0$ and $(M + m) = 1$.

• Applications Across Astronomy:

  • Exoplanetary Systems: Used to calculate orbits of planets around other stars where central star mass $M \neq 1\,\text{M}_{\odot}$ (e.g., stars with $2\,\text{M}_{\odot}$ or $0.5\,\text{M}_{\odot}$). Expressing variables in solar mass units ($\text{M}_{\odot}$), years ($\text{years}$), and astronomical units ($\text{AU}$) provides direct intuition by referencing Solar System baselines.

  • Galactic Dynamics: Used for stars orbiting the supermassive center of the Milky Way galaxy, operating with significantly larger mass scale values.

Newton's Laws of Motion and Universal Gravitation

• Historical Context:

  • During a university closure caused by an outbreak of the plague, Isaac Newton worked independently at home, developing calculus, classical mechanics, and the universal law of gravitation.

• Newton's Three Laws of Motion:

  1. First Law (Inertia): An object remains at rest or continues moving at a constant velocity in a straight line unless acted upon by a net external force that changes its speed or direction.

  2. Second Law ($F = M \times a$): Quantifies the relationship between force ($F$), mass ($M$), and acceleration ($a$).

     • Applies to linear motion and circular/orbital motion.

     • The Sun's gravitational force acts as a centripetal force on Earth, continuously redirecting Earth's velocity vector to maintain a closed orbit rather than allowing it to fly off linearly into space.

  3. Third Law (Action and Reaction): For every applied force (action), there is an equal magnitude force acting in the exact opposite direction (reaction).

• Newton's Universal Law of Gravitation:

  • Expressed mathematically as:         $F = \frac{G \times M_1 \times M_2}{r^2}$

  • $F$ = Gravitational force.

  • $G$ = Universal gravitational constant = $6.67 \times 10^{-11}\,\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$.

  • $M_1, M_2$ = Masses of the two interacting objects.

  • $r$ = Separation distance between the centers of the two objects.

• Proportional Scaling Rules:

  • Doubling mass $M_1$ ($2 \times M_1$) while keeping distance constant doubles the gravitational force ($2 \times F$).

  • Doubling mass $M_2$ ($2 \times M_2$) while keeping $M_1$ and distance constant doubles the gravitational force ($2 \times F$).

  • Doubling the distance $r$ ($2 \times r$) reduces the gravitational force by a factor of $4$ ($\frac{1}{4} \times F$) due to the inverse-square dependence ($r^2$).

Barycentric Orbits and Exoplanet Detection

• Equal and Opposite Gravitational Forces:

  • By Newton's Third Law, the gravitational force exerted by the Sun on the Earth is equal in magnitude to the gravitational force exerted by the Earth on the Sun.

  • By Newton's Second Law ($a = \frac{F}{M}$):

    • The Sun possesses a massive mass ($M_{\text{Sun}}$), resulting in a minute acceleration ($a_{\text{Sun}}$).

    • The Earth possesses a much smaller mass ($m_{\text{Earth}}$), resulting in a noticeably larger acceleration ($a_{\text{Earth}}$).

• Barycentric Motion:

  • Two gravitationally bound bodies orbit around their mutual center of mass, termed the barycenter.

  • Sun-Earth Barycenter: Because the Sun contains the vast majority of the system's mass, the Sun-Earth barycenter is located inside the interior volume of the Sun, close to its center.

  • Orbital Period Parity: The Sun's period of motion around the Sun-Earth barycenter is exactly $1\,\text{year}$, matching the Earth's orbital period.

  • Binary Star Systems: In a binary system containing two stars of equal mass, both stars orbit a central barycenter directly opposite each other with identical periods. In unequal-mass systems, forces remain equal and opposite, and orbital radii/accelerations differ, but the orbital periods must remain identical to keep the objects on opposite sides of the barycenter.

• Sun-Jupiter Barycentric Dynamics:

  • Jupiter is the dominant planetary mass in the Solar System.

  • The Sun-Jupiter barycenter is located just outside the physical surface of the Sun.

  • The Sun executes an orbital loop around this barycenter that is slightly larger than the diameter of the Sun itself, completing one full loop over an orbital period equal to Jupiter's orbital period (multiple years).

  • If Jupiter and all other planets were removed, leaving only Earth, the barycenter would shift back deep inside the Sun near its geometric center.

• Exoplanet Detection via Stellar Wobble:

  • The mutual gravitational interaction causes host stars to exhibit a subtle orbit or "wobble" around the center of mass shared with their orbiting planets.

  • Measuring this periodic stellar displacement enables astronomers to detect non-luminous exoplanets and derive their orbital periods and masses.

Course Logistics

• Canvas Submission Requirement:

  • Task: Friday upload under Canvas modules requiring entry of calculated height in non-standard units.

  • Access Code: units (case-sensitive, lowercase).

• Homework Deadline:

  • Homework assignment due at 5:00 PM on the day assigned.