Question

Suppose that we decide to send a spacecraft to Neptune, using the simple transfer described in Example 8.6 (page 318). The craft starts in a circular orbit close to the earth (radius 1 AU or astronomical unit) and is to end up in a circular orbit near Neptune (radius about \(30 \mathrm{AU}\) ). Use Kepler's third law to show that the transfer will take about 31 years. (In practice we can do a lot better than this by arranging that the craft gets a gravitational boost as it passes Jupiter.)

Short answer

The transfer takes about 31 years using Kepler's Third Law.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law states that the square of the period of orbit (T) of a planet around the Sun is proportional to the cube of the semi-major axis (a) of its orbit. Mathematically, this is expressed as \( T^2 \propto a^3 \). For transfer orbits, a similar setup can be used.

Define the Semi-Major Axis for the Transfer Orbit

The semi-major axis \( a \) for the spacecraft's transfer orbit is the average of the Earth's orbit radius and Neptune's orbit radius. Thus, \( a = \frac{1 + 30}{2} \mathrm{AU} = 15.5 \mathrm{AU} \).

Apply Kepler's Third Law

For solar system objects, we use the proportional relation \( T^2 = a^3 \), with \( T \) in years when \( a \) is in astronomical units. Substitute into this relation: \( T^2 = (15.5)^3 \).

Calculate Duration of Transfer Orbit

Solve \( T^2 = (15.5)^3 \), so \( T = \sqrt{15.5^3} \). Compute \( 15.5^3 = 3723.875 \), then \( T = \sqrt{3723.875} \approx 61 \) years (for the full round trip).

Determine One-Way Transfer Time

The calculated 61 years is for the full orbit around the Sun. Therefore, one-way to Neptune will take \( \frac{61}{2} = 30.5 \) years, approximately 31 years.

Key concepts

transfer orbit

A transfer orbit is a trajectory followed by a spacecraft that allows it to move between two celestial bodies in space, such as planets in the solar system. This is essential when sending missions to distant planets or moons. In our exercise, the spacecraft begins its journey in Earth's orbit and aims to reach Neptune's orbit. To achieve this, the spacecraft will leverage what is known as a Hohmann Transfer Orbit. This type of orbit is an efficient elliptical path that minimizes the fuel needed for the journey by aligning with the gravitational pull of the Sun.

Specifically, for the transfer from Earth to Neptune, the spacecraft initially travels along an elliptical orbit that has its perihelion (closest point to the Sun) at Earth's orbit and its aphelion (farthest point from the Sun) at Neptune's orbit. This makes the path energy-efficient, as it uses the least amount of propulsion.

There are a few key aspects involved in designing a transfer orbit:

• The timing of the launch has to be precise to ensure that when the spacecraft arrives at its aphelion, Neptune is positioned where the spacecraft can enter its orbit.
• Adjustments can be made for the spacecraft to receive gravitational boosts from other planets, like Jupiter, to reduce travel time.

semi-major axis

The semi-major axis is a crucial parameter in understanding orbits, particularly elliptical ones. It represents the longest diameter of an ellipse, extending from the center through the longest part of the ellipse to the perimeter, essentially half the longest diameter. In the context of celestial mechanics and Kepler's laws, it provides a measure of the size of an orbit.

In our exercise, we determined the semi-major axis of the transfer orbit by averaging the distances of Earth's and Neptune's orbits from the Sun. Since Earth's orbit is 1 astronomical unit (AU) from the Sun and Neptune's orbit is roughly 30 AU from the Sun, their average gives us a semi-major axis of 15.5 AU for the transfer orbit.

Why is this important? According to Kepler's Third Law, the time it takes for a body to orbit the Sun (orbital period) is closely related to the semi-major axis of its orbit. By knowing the semi-major axis, we can predict the time period necessary for the spacecraft to travel from Earth to Neptune. Essentially, a longer semi-major axis results in a longer orbital period.

Knowing about the semi-major axis helps us use Kepler's Third Law to forecast travel duration and optimize mission planning by adjusting trajectory before launch.

astronomical unit

An astronomical unit (AU) is a unit of length used in astronomy to describe distances within our solar system. It is approximately equal to the average distance from the Earth to the Sun, which is about 149.6 million kilometers or roughly 93 million miles.

The concept of an astronomical unit makes it easier to express and compare vast distances between planets and other celestial bodies in the solar system without using cumbersome large numbers.

In this exercise, we used astronomical units to measure the spacecraft's starting and target orbits, specifically 1 AU for Earth and around 30 AU for Neptune. This measurement simplifies the application of Kepler's Third Law in computing the semi-major axis and orbital period since AU offers a consistent and relatable scale.

When it comes to understanding space travel and telescope observations, astronomical units provide a helpful reference for interpreting distances. For students and astronomers alike, it offers a convenient way to visualize, calculate, and communicate the vast expanses of space with greater clarity and context.