Orbits - Answers to Q.

  1. Kepler's Three Laws of Planetary Motion:

    • First Law (Law of Ellipses): The path of the planets around the sun is an ellipse, with the sun at one focus.

    • Second Law (Law of Equal Areas): A line segment joining a planet and the sun sweeps out equal areas during equal intervals of time.

    • Third Law (Law of Harmonies): The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit (T² ∝ r³).

  2. Newton's Law of Gravitation and Planetary Motion:

    • Newton's law explains that every mass attracts every other mass with a force given by F = G M1M2 / r². This gravitational force is what keeps planets in orbit around the sun and satellites in orbit around planets.

    • Example: The Earth orbits the Sun because the gravitational pull provided by the Sun (M1) acting on Earth (M2) counterbalances the centrifugal force due to its orbital motion.

  3. Deriving Kepler's 3rd Law for Circular Orbits:

    • Start from Newton's law of gravity: F = G M1M2 / r² and the formula for centripetal acceleration: F = (M2 v²) / r. For a circular orbit of radius r, the gravitational force provides the centripetal acceleration needed for the planet's orbit. Setting the two forces equal gives G M1M2 / r² = M2 v² / r. Simplifying leads to v² = G M1 / r.

    • The orbital period T is related to velocity and radius as v = 2πr / T. Substituting this in gives T² = (4π² / GM1) r³, leading to Kepler's 3rd Law: T² ∝ r³.

  4. Using Orbital Motion Data to Calculate Mass:

    • The mass of a central object can be determined using the orbital period (T) and the radius (r) of the orbiting body. By rearranging Kepler's Third Law, M1 can be found: M1 = (4π²r³) / (GT²).

  5. Orbital Speeds and Dark Matter:

    • In spiral galaxies, the orbital speeds of stars remain constant at greater distances from the center, which contradicts expectations based on visible mass. This suggests the presence of dark matter, which provides additional gravitational force that is not visible.

  6. Higgs Boson and Dark Matter:

    • The relationship between the Higgs boson and dark matter is still under investigation. The Higgs boson gives mass to elementary particles, but dark matter does not seem to interact with electromagnetic forces, making its relation to the Higgs complex and not fully understood.

  7. Center of Mass and Mutual Orbital Period:

    • The center of mass (CM) of two spherically symmetric objects can be found using CM = (m1 * x1 + m2 * x2) / (m1 + m2) where m1 and m2 are the masses and x1 and x2 are their positions. For mutual orbital period in circular orbits, the formula derived will involve the distances from the CM and their total mass using Kepler's laws.

  8. Doppler Relationship:

    • The Doppler relationship states that the change in wavelength (Δλ) relative to the original wavelength (λ) is proportional to the velocity (v) of the source relative to the observer divided by the speed of light (c): Δλ / λ = v / c.

  9. Determining Radial Velocity:

    • A star's radial velocity is determined by measuring the shift in the wavelength of spectral lines due to the Doppler effect. If the lines are blue-shifted, the star is moving towards us, and if red-shifted, it is moving away.

  10. Variation of Radial Velocities in a Binary System:

    • In a double system, the periodic variation in radial velocities of a star and an orbiting exoplanet can be used to derive their masses via Kepler's Third Law combined with Doppler shifts observed from Earth. The mass ratio can be determined from the amplitude of the radial velocity curves.

  11. Hubble Constant (H0):

    • The Hubble constant relates the radial velocity (v) of galaxies to their distance (D), defined by v = H0D. It describes the rate at which the universe is expanding.

  12. Age of the Universe:

    • The inverse of the Hubble constant (1 / H0) approximates the age of the universe because it represents the time taken for the universe to expand to its current size assuming a constant rate of expansion.

  13. Critical Density Equation:

    • The equation ρc = 3H0² / (8πG) for critical density can be derived using conservation of energy principles. It relates the density required for the universe to be flat with the Hubble parameter, indicating how much matter/energy density is needed to balance the expansion of the universe.