Orbital Mechanics and Ellipses Quiz

Lecture 3: Binary Stars

• Lecture was on binary stars, part of the "Observing the Universe" course
• The lecturer was Prof. Evan Keane
• The date of the lecture was Thursday 23rd January 2025 at 12:00
• Lecture 2 covered spectral classification of stars, spectral line origins, relevant physical processes, and the Hertzsprung-Russell diagram.
• Lecture 3 goal: Understand binarity manifestations in stars and what can be learned from quantifying binarity.
• Lecture 3 outline: Binary parameters, Kepler's laws, types of observed binaries, Doppler effect, stellar masses.

L2 Recap

• Spectral classification of stars
• Origin of spectral lines
• Physical processes relevant to spectral lines
• The Hertzsprung-Russell diagram

L3 Learning Outcomes

• Understanding the manifestations of binarity in stars
• Understanding what can be learned by quantifying binary systems
• Binary parameters and Kepler's laws
• Types of observed binaries
• Doppler effect
• Stellar masses

Aside: Ellipses

• Equation of an ellipse: x²/a² + y²/b² = 1
• Area of an ellipse: πab
• For a circle, a = b = r (center of circle is the focal point)
• Distance from origin to focus: c = ae
• For any point on ellipse: |F₁P| + |F₂P| = const = 2a

Binary Parameters

• An orbit is defined by 6 orbital elements
• Closed orbits are elliptical and defined by 2 parameters:
  • a: Semi-major axis (size)
  • e: Eccentricity (0 = circle, 0<e<1 = ellipse, e=1 = parabola, e>1 = hyperbola)
  • b: Semi-minor axis (need either (a, b) or (a, e) to define ellipse shape)
• i: Orbital plane's inclination with respect to the observer
• Ω: Longitude of ascending node, measured in the observer's plane
• ν: True anomaly (angle defining the epoch of the orbital body during observation, cyclic in orbital period Pb)
• Argument of periapsis (ω) and longitude of ascending node (Ω) define the orbital plane orientation
• Terms like periapsis, periastron, perihelion, and perigee refer to the closest point in a system's orbit.

Jargon Buster

• Periapsis, periastron, perihelion, and perigee are all geometrically the same
• Terms differ based on the specific system in question (e.g., perihelion refers to the Earth's closest point to the Sun)
• Apapsis, apastron, aphelion, and apogee refer to the furthest point in an orbit.

Theories of Gravity

• Orbital elements are independently measurable for any observed binary system in the sky
• Theories of gravity define how interactions/motion happen
• The relationships between orbital elements, physical constants (G), and charges (like masses) can be predicted by a complete theory of gravity.

Kepler Orbits

• Kepler's laws for orbits, based on Newtonian gravity:
  • K1: Orbits are ellipses
  • K2: dA/dt = const
  • K3: P² = 4π²a³/GM
  • (where M = m₁ + m₂)

Theories of Gravity (cont.)

• Orbital elements can change with time, with additional parameters such as w(t) and Pb(t).
• These can also be used to test different gravitational theories, independently of the specific model of the system.

Doppler Effect

• The Doppler effect is a change in frequency of a wave (or other periodic event) for an observer moving relative to its source.
• Δν/ν = v/c. (where Δν is the change in frequency, ν is the emitted frequency, v is the relative velocity between source and observer and c is the speed of light)

Stellar Masses

• Using methods described, stellar masses can be determined
• Applicability is not limited to stars; also applies to exoplanets.

Stellar Properties

• Combining information from L1-L3 allows a lot to be learned about stars.
• Measure stellar flux and distance (parallax -> bolometric luminosity)
• Measure spectral features (spectral lines, shape) to determine stellar temperature and composition.
• The Stefan-Boltzmann relation is used to calculate stellar radii
• Binary systems enable mass determination

Mass-Luminosity Relation

• L ∝ M^3.5 (Luminosity is proportional to the mass raised to 3.5)
• Observationally derived
• Can be related to internal star chemistry and structure

Stellar Ages

• Calculating time-scales like timescale = (anything)/(d(anything)/dt)
• Age ~ E/L (Energy divided by Luminosity) to determine how long a process can fuel a star
• Gravitational contraction can be used to estimate the Kelvin-Helmholtz timescale (~20 million years for Sun).
• Nuclear fusion is required for long term sustained energy in the Sun.

Problems 3.1, 3.2, 3.3, 3.4

• Questions regarding binary systems, including angles of inclination, size of orbits, relativistic effects, Doppler shifts, and stellar properties.