Kepler's First Law: Videos & Practice Problems

Kepler's first law states that all orbits are elliptical, with the sun at one focus. The major axis length is represented as \(2a\), where \(a\) is the semi-major axis. The closest point in orbit is called perihelion (\(R_P\)), and the farthest is aphelion (\(R_A\)), satisfying the equation \(R_A + R_P = 2a\). Eccentricity (\(e\)) measures orbit shape, with values near 0 indicating circular orbits and values near 1 indicating highly elliptical orbits. The relationships are given by \(R_A = a(1 + e)\) and \(R_P = a(1 - e)\).

Kepler's First Law

Kepler's first law states that all planetary orbits are elliptical, with the sun located at one of the two foci of the ellipse. This means that even circular orbits are a special case of ellipses. The major axis of the ellipse is the longer axis, divided into two halves, each represented by the variable a, making the total length of the major axis equal to 2a.

In an elliptical orbit, the closest point to the sun is called the periapsis (or perihelion for orbits around the sun), denoted as R_P, while the farthest point is called the aphelion (or apoapsis), denoted as R_A. The relationship between these distances and the semi-major axis can be expressed as:

R_A + R_P = 2a

From this, the semi-major axis a can be calculated as:

a = \frac{R_A + R_P}{2}

The minor axis, which is the shorter axis of the ellipse, has a length of 2b, where b is half of the minor axis.

Kepler also introduced the concept of eccentricity, denoted as e, which measures the deviation of an orbit from being circular. Eccentricity values range from 0 to 1, where values close to 0 indicate nearly circular orbits (like those of most planets) and values close to 1 indicate highly elliptical orbits (like those of comets). The eccentricity can be calculated using the following relationships:

R_A = a(1 + e)

R_P = a(1 - e)

In circular orbits, the distances R_A and R_P are nearly equal, resulting in an eccentricity of approximately 0.

To illustrate these concepts, consider Earth's orbit. The closest distance to the sun (perihelion) is approximately 1.471 × 10¹¹ m, and the farthest distance (aphelion) is about 1.521 × 10¹¹ m. The semi-major axis can be calculated as:

a = \frac{1.471 × 10¹¹ + 1.521 × 10¹¹}{2} = 1.496 × 10¹¹ m

Next, to find the eccentricity, we can use the aphelion equation:

R_A = a(1 + e)

1.521 × 10¹¹ = 1.496 × 10¹¹(1 + e)

By simplifying, we find:

1.017 = 1 + e

e = 0.017

This small eccentricity indicates that Earth's orbit is very close to circular, which is consistent with the orbits of most planets in our solar system.

Pluto's orbit is the most eccentric of the 9 large objects in our solar system, with e = 0.25. The total distance from Pluto's closest to farthest point from the Sun is 1.18×10¹³m. a) How close does it get to the Sun? b) How far does it get from the Sun?

• A. (a) R_p=4.4×10¹² m;
  (b) R_a=7.4×10¹² m
• B. (a) R_p=8.8×10¹² m;
  (b) R_a=1.5×10¹³ m
• C. (a) R_p=1.8×10¹³ m;
  (b) R_a=3.0×10¹³ m

Answer: (A)

Here’s what students ask on this topic:

What is Kepler's First Law and how does it describe planetary orbits?

Kepler's First Law, also known as the Law of Ellipses, states that all planets move in elliptical orbits with the Sun at one of the two foci. This means that the path of a planet around the Sun is not a perfect circle but an elongated circle, or ellipse. The Sun is located at one focus of the ellipse, while the other focus is an empty point in space. This law helps explain the varying distances between a planet and the Sun during its orbit, leading to different orbital speeds. The closest point to the Sun is called the perihelion, and the farthest point is the aphelion. Understanding this law is crucial for comprehending the dynamics of planetary motion in our solar system.

How do you calculate the semi-major axis of an elliptical orbit?

The semi-major axis of an elliptical orbit is a crucial parameter that represents half of the longest diameter of the ellipse. It can be calculated using the formula: $a=\frac{(R_A+R_P)}{2}$, where $R_A$ is the distance at aphelion (farthest point from the Sun) and $R_P$ is the distance at perihelion (closest point to the Sun). By averaging these two distances, you obtain the semi-major axis, which is a fundamental element in determining the size and shape of the orbit.

What is eccentricity in the context of Kepler's First Law?

Eccentricity, denoted by the symbol $e$, is a measure of how much an orbit deviates from being circular. In the context of Kepler's First Law, eccentricity ranges from 0 to 1. An eccentricity of 0 indicates a perfect circle, while an eccentricity close to 1 indicates a highly elongated ellipse. The formula for eccentricity is derived from the semi-major axis and the distances at perihelion and aphelion: $R_A=a(1+e)$ and $R_P=a(1-e)$. Eccentricity helps describe the shape of the orbit and is essential for understanding the orbital dynamics of celestial bodies.

How does Kepler's First Law apply to Earth's orbit around the Sun?

Kepler's First Law applies to Earth's orbit by describing it as an ellipse with the Sun at one focus. Although Earth's orbit is nearly circular, it is technically an ellipse with a small eccentricity of approximately 0.017. This means that the distance between Earth and the Sun varies slightly throughout the year. The closest point, called perihelion, occurs in early January, while the farthest point, called aphelion, occurs in early July. The semi-major axis of Earth's orbit is about 149.6 million kilometers, which is the average distance from the Earth to the Sun. Understanding this law helps explain seasonal variations and the slight changes in solar energy received by Earth.

What are the key differences between the major and minor axes in an elliptical orbit?

In an elliptical orbit, the major and minor axes are the longest and shortest diameters, respectively. The major axis runs through both foci of the ellipse and is divided into two equal halves, each of length $a$, making the total length $2a$. The minor axis is perpendicular to the major axis at the center of the ellipse and is divided into two equal halves, each of length $b$, making the total length $2b$. The major axis determines the overall size of the orbit, while the minor axis indicates the degree of elongation. Together, they define the shape and dimensions of the elliptical path.