Comets move around the sun in very elliptical orbits. At its closet approach, in 1986, Comet Halley was 8.79 x 10⁷ km from the sun and moving with a speed of 54.6 km/s. What was the comet’s speed when it crossed Neptune’s orbit in 2006?

Verified step by step guidance

1. Step 1: Recognize that this problem involves the conservation of energy. The total mechanical energy of the comet (kinetic energy + gravitational potential energy) remains constant throughout its orbit, assuming no external forces act on it.
2. Step 2: Write the expression for the total mechanical energy at the closest approach (perihelion): \( E = \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} \), where \( m \) is the mass of the comet, \( v_1 \) is the speed at perihelion (54.6 km/s), \( r_1 \) is the distance at perihelion (8.79 × 10⁷ km), \( G \) is the gravitational constant, and \( M \) is the mass of the Sun.
3. Step 3: Write the expression for the total mechanical energy at Neptune’s orbit (aphelion): \( E = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \), where \( v_2 \) is the speed at Neptune’s orbit (what we are solving for) and \( r_2 \) is the distance of Neptune’s orbit from the Sun (approximately 4.50 × 10⁹ km).
4. Step 4: Set the total mechanical energy at perihelion equal to the total mechanical energy at aphelion, since energy is conserved: \( \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \). Simplify by canceling \( m \) from all terms.
5. Step 5: Rearrange the equation to solve for \( v_2 \): \( v_2 = \sqrt{v_1^2 + 2 G M \left( \frac{1}{r_1} - \frac{1}{r_2} \right)} \). Substitute the known values for \( v_1 \), \( r_1 \), \( r_2 \), \( G \) (6.674 × 10⁻¹¹ N·m²/kg²), and \( M \) (1.989 × 10³⁰ kg) to calculate \( v_2 \).

Key Concepts

Elliptical Orbits

Elliptical orbits describe the path of celestial bodies, such as comets, around a focal point, typically a star like the sun. According to Kepler's laws of planetary motion, these orbits are elongated circles, with varying speeds depending on the distance from the sun. As a comet approaches the sun, it accelerates due to gravitational attraction, and as it moves away, it decelerates.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. For a comet, this means that its kinetic energy (related to its speed) and potential energy (related to its distance from the sun) will interchange as it moves along its orbit. This principle allows us to calculate the comet's speed at different points in its orbit by equating the energy at those points.

Gravitational Influence

Gravitational influence refers to the effect that a massive body, like the sun, has on other objects in its vicinity. The strength of this gravitational pull decreases with distance, affecting the speed and trajectory of orbiting bodies. In the case of Comet Halley, its speed changes as it moves through different regions of the solar system, influenced by the sun's gravity and the gravitational effects of other planets.