Question

Halley's comet orbits the Sun with a period of 76.2 yr. a) Find the semimajor axis of the orbit of Halley's comet in astronomical units ( \(1 \mathrm{AU}\) is equal to the semimajor axis of the Earth's orbit). b) If Halley's comet is \(0.56 \mathrm{AU}\) from the Sun at perihelion, what is its maximum distance from the Sun, and what is the eccentricity of its orbit?

Short answer

Answer: The semimajor axis of Halley's comet's orbit is approximately 17.9 AU, the maximum distance (aphelion) from the Sun is approximately 35.24 AU, and the eccentricity of the orbit is approximately 0.97.

Step-by-step solution

Use Kepler's third law to find the semimajor axis of Halley's comet's orbit

Kepler's third law states that \((\frac{a}{1 AU})^3 = (\frac{T}{1 yr})^2\), where \(a\) is the semimajor axis of the orbit, \(T\) is the period, \(1 AU\) is the astronomical unit (the semimajor axis of Earth's orbit around the Sun), and \(1 yr\) is one Earth year. Here, \(T = 76.2 yr\). Substituting the values and solving for \(a\), we get: \((\frac{a}{1 AU})^3 = (\frac{76.2 yr}{1 yr})^2\) \(a^3 = (76.2^2) (1 AU)^3\) \(a = \sqrt[3]{(76.2)^2 (1 AU)^3}\)

Calculate the semimajor axis of Halley's comet's orbit

Now, we can plug in the values and solve for the semimajor axis \(a\): \(a = \sqrt[3]{(76.2)^2 (1 AU)^3}\) \(a = \sqrt[3]{5800.84 (1 AU)^3}\) \(a \approx 17.9 AU\) So, the semimajor axis of Halley's comet is approximately \(17.9 AU\).

Find the maximum distance (aphelion) of Halley's comet from the Sun

We are given that the distance from the Sun at perihelion is \(0.56 AU\). Let \(q\) be the perihelion distance and \(Q\) be the aphelion distance. We know that \(a = \frac{q+Q}{2}\), so we can solve for \(Q\): \(Q = 2a - q\) \(Q = 2(17.9 AU) - 0.56 AU\) \(Q \approx 35.24 AU\) The maximum distance (aphelion) of Halley's comet from the Sun is approximately \(35.24 AU\).

Find the eccentricity of Halley's comet's orbit

To find the eccentricity \(e\) of Halley's comet's orbit, we use the formula: \(e = \frac{Q-q}{Q+q}\) Plugging in the values for \(Q\) and \(q\), we get: \(e = \frac{35.24 AU - 0.56 AU}{35.24 AU + 0.56 AU}\) \(e = \frac{34.68 AU}{35.8 AU}\) \(e \approx 0.97\) The eccentricity of Halley's comet's orbit is approximately \(0.97\).

Key concepts

Semimajor Axis

The semimajor axis of an orbit in astronomy is a crucial concept as it defines the size of the orbit. For elliptical orbits, which most bodies in our solar system follow, the semimajor axis is half of the longest diameter of the ellipse. This is a central parameter when it comes to calculating various orbital properties, including the orbital period and the distances at various points around the orbit.
Kepler's third law provides a simple relationship between the semimajor axis and the orbital period. It states that the square of the period of orbit (T^2) is proportional to the cube of the semimajor axis (a^3). For orbits with the period measured in Earth years and the semimajor axis in astronomical units (AU), Kepler's third law simplifies and becomes\(\left(\frac{a}{1 AU}\right)^3 = \left(\frac{T}{1 yr}\right)^2\).
In the context of Halley's comet, which has a period of 76.2 years, applying Kepler's law gives us a semimajor axis of approximately 17.9 AU.

Eccentricity of Orbit

Eccentricity measures how much an orbit deviates from being a perfect circle. It is a dimensionless parameter that ranges from 0 to 1, where 0 indicates a circular orbit and values closer to 1 indicate more elongated orbits. Knowing the eccentricity helps us understand the shape and dynamics of an orbit.
The formula to calculate eccentricity (e) is derived from the distances of closest approach (perihelion, q) and furthest point (aphelion, Q) from the Sun:\(e = \frac{Q-q}{Q+q}\)
For Halley's comet, where the perihelion distance is 0.56 AU and the aphelion distance is approximately35.24 AU, we find the eccentricity to be about 0.97. This high eccentricity suggests that Halley's comet travels a highly elongated path around the Sun, spending much of its time far away from the Sun, only coming close every 76.2 years.

Astronomical Units

Astronomical Units (AU) are a standard unit of measurement used in astronomy to describe distances within our solar system. One AU is defined as the average distance between the Earth and the Sun, approximately 149,597,870.7 kilometers. This unit allows astronomers to easily express and compare distances in the vast expanses of space.
Using AU simplifies many calculations and allows for easier comprehension of relative distances. When the semimajor axis of an orbit is expressed in terms of AU, it relates directly to Earth’s orbit size, offering an intuitive comparison.
In the study of Halley's comet, its semimajor axis being 17.9 AU means it is 17.9 times further from the Sun than the average Earth-Sun distance, indicating extensive travel through space both near and far from our planetary neighborhood.