4.4: Orbits and Rotation

Movement of Objects in the Solar System

Now that we know what is in the Solar System, we will examine how those objects move. Still images of objects in the Solar System do not reflect the fact that they are all constantly moving. Even the Sun is in motion as it orbits around the center of the Milky Way galaxy. To keep things simple, in this textbook we are going to the examine the motions of objects in the Solar System relative to the Sun. This perspective is also known as using the Sun as our point of reference, or setting an imaginary reference frame on the Sun as stationary. For example, if we used this reference frame to plot the Sun's coordinates on an x-y graph, it would be located at (0,0). Our reference frame then moves with the Sun, meaning that the speed of the Sun inside the reference frame is zero.    

There are several basic motions of objects in the Solar System. Revolution is one object orbiting another object. Earth revolves around, or orbits, the Sun approximately every 365.24 Earth days (1 Earth year), while Mars orbits the Sun every 687 Earth days (1 Martian year). Rotation is the spinning of an object around its axis. Earth rotates once every 24 hours, whereas Mars rotates once every 24.6 hours. 

Orbits

When originally conceived, the heliocentric model assumed that all planets orbit around the Sun along the ecliptic and all moving in the same direction. To be able to accurately predict the locations of the planets over time, astronomers needed to fully understand the orbits of the planets. Understanding the orbits required detailed observations over long periods of time. These observations could be used to refine and make improvements to the heliocentric model.

In the second half of the 1500s and early 1600s, Tycho Brahe, established an observatory where he was able to make a continuous record of the positions of the Sun, Moon, and planets for almost 20 years. His extensive and precise observations enabled him to note that the positions of the planets varied from those given in published tables, which were based on the geocentric model. While Brahe was very good at measuring and recording the positions of objects in the sky, he did not have the mathematical training to create a model.  Fortunately, he met the German mathematician, Johannes Kepler. Using Brahe's observational data, Kepler developed a series of principles, now known as Kepler’s three laws, which described the behavior of planets based on their paths through space.

Circles vs Ellipses  

Kepler initially assumed that the orbits of planets were circles, but doing so did not allow him to find orbits that were consistent with Brahe’s observations. Working with the data for Mars, he eventually discovered that the orbit of that planet had the shape of a somewhat flattened circle, or ellipse.

What is an ellipse? In a circle, the distance from the center to anywhere on the circle is exactly the same. In an ellipse, the sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same. These two special points inside the ellipse are called its foci (singular: focus), a word invented for this purpose by Kepler.

To draw an ellipse, as shown in Figure \(\PageIndex{1}\), we can wrap the ends of a loop of string around two pegs pushed through a sheet of paper into a drawing board. If we push a marker against the string, and then slide the marker against the string all around the pegs, the resulting shape is an ellipse. At any point where the marker may be, the sum of the distances from the marker to the two pegs is a constant length—the length of the string. The pegs are at the two foci of the ellipse. 

Figure \(\PageIndex{1}\): Drawing an Ellipse. (a) We can draw an ellipse by pushing two pegs into a piece of paper on a drawing board, and then looping a string around the pegs. The two pegs are called the foci (singular focus) of the ellipse. Stretch the string tight using a marker, and then move the marker around the tacks. The length of the string remains the same, so that the sum of the distances from any point on the ellipse to the foci is always constant.  ("Ellipse drawn using a loop of string," John Richfield, CC0, via Wikimedia Commons)

The widest diameter of the ellipse is called its major axis. Half this distance—that is, the distance from the center of the ellipse to one end—is the semi-major axis. In data tables, semi-major axis, represented by the variable a, is used to represent the size of an orbit. For example, the semi-major axis of the orbit of Mars, which is also the planet’s average distance from the Sun, is 228 million kilometers. 

Figure \(\PageIndex{2}\): Parts of an Ellipse. The center of this ellipse is labelled with the letter C, while F₁ and F₂ are the locations of the two foci. The semi-major axis, red, labbeled a, is half of the longest width of the ellipse. The semi-minor axis, blue line labeled b, is half of the shortest width of the ellipse.  ("Ellipse semi-major and minor axes", M. W. Toews, CC0, via Wikimedia Commons)

The semi-major axis, a, is the average of the distance of a planet from the Sun. However, instead of being at the center of the ellipse, or at letter C in Figure \(\PageIndex{2}\), the Sun is at one of the two foci while the other focus is empty. As the planet orbits the Sun, the distance between the planet and the Sun changes. The general term for the shortest distance between one object orbiting another is periapsis and the longest distance between two objects is apoapsis. Note that the prefixes for these terms are peri, near in Greek and apo, away from in Greek. For the orbits of the planets around the Sun, Kepler named the position where the planet is closest to the Sun, perihelion, while the position when the planet is farthest from the Sun is called aphelion, Figure \(\PageIndex{3}\).  Remember that Helios is the Greek god of the Sun. There are various terms for orbital position that follow this pattern and we will be using them throughout the textbook. For example, when the Moon is closest to the Earth it is at perigee and when it is farthest from the Earth, it is at apogee.  The terms apogee and perigee also apply to human-built satellites in orbit around the Earth.

Figure \(\PageIndex{3}\): Perihelion and Aphelion. This image demonstrates the position of aphelion, when a planet is farthest from the Sun, and the position of perihelion when a planet is closest to the Sun. The semi-major axis of the planet is shown in red. Note that the aphelion is longer than the semi-major axis, while the perihelion is shorter than the semi-major axis. Keep in mind that the planets have orbits that are much more circular than this example. Therefore, the difference between their aphelion and perihelion is much smaller. ("Perihelion aphelion semimajor axis", Maxmath12, CC0, via Wikimedia Commons)

The shape, or roundness, of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse. Figure \(\PageIndex{4}\) includes five shapes.  If the foci (or pegs) are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semi-major axis is equal to the radius.  The yellow ellipse has two foci - the red dot and the yellow dot.  The green, aqua, and blue dots are at progressively farther distances from the red dot.  This means that as the eccentricity increases, the ellipse becomes wider and wider.  However, the eccentricity can't be larger than one because that would be like drawing an ellipse with a loop of string that barely fit over the two pegs in Figure \(\PageIndex{1}\). We would end up with a straight line between the foci. This means that all ellipses have an eccentricity between zero and one. The higher the eccentricity, the larger the semi-major axis is compared to the semi-minor axis, creating a wider ellipse.

Figure \(\PageIndex{4}\): Ellipses and Eccentricity. The red circle in this figure has a red dot in the middle. That same red dot is one of the foci of an ellipse while the dot of matching color is the other foci for the four ellipses that are yellow, green, aqua and blue. The blue dot is the largest distance from the red dot and the blue ellipse is the widest. ("Ellipse set 2", Public Domain; Orion 8, CC0, via Wikimedia Commons)

The size and shape of an ellipse are completely specified by its semi-major axis and its eccentricity. Using Brahe’s data, Kepler found that Mars has an elliptical orbit, with the Sun at one focus and the other focus is empty. The eccentricity of the orbit of Mars is only about 0.1; its orbit, drawn to scale, would be practically indistinguishable from a circle. However, this difference turned out to be critical for understanding planetary motions. Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses. Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.

Equal Areas, Equal Time

Kepler’s second law deals with the speed with which each planet moves along its ellipse, also known as its orbital speed. Working with Brahe’s observations of Mars, Kepler discovered that the planet speeds up as it comes closer to the Sun and slows down as it pulls away from the Sun. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun (positions 1 and 2 in Figure 3.5), the elastic line is not stretched as much, and the planet moves rapidly. Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves (the colored regions in our figure). Kepler found that in equal intervals of time (t), the areas swept out in space by this imaginary line are always equal; that is, the area of the region B from 1 to 2 is the same as that of region A from 3 to 4. If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But, as Kepler discovered, the speed of a planet orbiting its star tends to vary because the orbit is often elliptical. Specifically, the planet moves the fastest near the Sun, and the slowest when it is far away. If you want to see Kepler's second law in action, you can try the simulation below. 

Figure \(\PageIndex{5}\): The Law of Equal Areas. The orbital speed of a planet traveling around the Sun varies in such a way that in equal intervals of time (t), a line between the Sun and a planet sweeps out equal areas (A and B). This means that the planet speeds up when it is near the Sun and slows down when it is far from the Sun. Note that the eccentricities of the planets’ orbits in our Solar System are substantially less than shown here. (Openstax Astronomy 2nd ed.)

 

Orbital Periods

Kepler’s first two laws of planetary motion describe the shape of a planet’s orbit and allow us to calculate the speed of its motion at any point in the orbit. Kepler was pleased to have discovered such fundamental rules, but they did not satisfy his quest to fully understand planetary motions. He wanted to know why the orbits of the planets were spaced as they are and to find a mathematical pattern in their movements—a “harmony of the spheres” as he called it. For many years he worked to discover mathematical relationships governing planetary spacing and the time each planet took to go around the Sun.

In 1619, Kepler discovered a basic relationship to relate the planets’ orbits to their relative distances from the Sun. We define a planet’s orbital period, (P), as the time it takes a planet to travel once around the Sun. Also, recall that a planet’s semimajor axis, a, is equal to its average distance from the Sun. The relationship, now known as Kepler’s third law, says that a planet’s orbital period squared is proportional to the semimajor axis of its orbit cubed.

When P (the orbital period) is measured in years and a is expressed in a quantity known as an astronomical unit (AU), the two sides of the formula are not only proportional, but actually equal. One AU is the average distance between Earth and the Sun and is approximately equal to $1.5\times {10}^8$ kilometers. In these units:

P² = a³       

where P is the period, or the time it takes for an object to complete one orbit in Earth years, and a is the semi-major axis of the object's orbit in AU.

Kepler’s third law applies to all objects orbiting the Sun, including Earth, and provides a means for calculating their relative distances from the Sun from the time they take to orbit. Let’s look at a specific example to illustrate how useful Kepler’s third law is.

For instance, suppose you time how long Mars takes to go around the Sun (in Earth years). Kepler’s third law can then be used to calculate Mars’ average distance from the Sun. Mars’ orbital period (1.88 Earth years) squared, or $P^2$, is ${1.88}^2=3.53$, and according to the equation for Kepler’s third law, this equals the cube of its semi-major axis, or $a^3$. So what number must be cubed to give 3.53? Using a calculator, the answer is $\mathrm{1.52 }$$(\text{since}1.52\times 1.52\times 1.52=3.53)$. Mars’ semi-major axis in astronomical units must be 1.52 AU. Interpreting the actual meaning of these numbers: to go around the Sun in a little less than two years, Mars must be about 1.5 times greater distance from the Sun as Earth is. 

Summary of Kepler's Laws

Kepler’s First Law

The planets travel around the Sun in elliptical orbits. Copernicus thought the planets moved in perfect circles, whereas Kepler defined these as ellipses, based on Brahe’s data.

Kepler’s Second Law

As a planet orbits the Sun, it sweeps out equal areas of its ellipse in equal periods of time. The closer the planet to the Sun (or its star), the faster it moves.

Kepler’s Second Law is stated as:

 

Where:

• v is the orbiting object’s velocity
• a is the semimajor axis of the object’s orbit
• P is the sidereal period of revolution
• r is the distance between the orbiting object and the body being orbited, such as Earth orbiting the Sun, or the Moon orbiting Earth

Kepler’s Third law

A relationship exists between the planet’s period and its distance from the Sun.

Kepler’s Third Law is stated as:

Where:

• a is the orbiting object’s semimajor axis
• P is the orbiting object’s period to orbit
• r is a constant, referred to as Kepler’s constant

 

Kepler’s three laws of planetary motion can be summarized as follows:

• Kepler’s first law: Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
• Kepler’s second law: The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
• Kepler’s third law: The square of a planet’s orbital period is directly proportional to the cube of the semi-major axis of its orbit.

Kepler’s three laws provided a precise geometric description of planetary motion within the framework of the Copernican system. With these tools, it was possible to calculate planetary positions with greatly improved precision. Still, Kepler’s laws are purely descriptive, meaning they do not help us understand what forces of nature constrain the planets to follow this particular set of rules. That step was left to Isaac Newton. Mention Kepler simulation at bottom of page

Physics of Orbits

Newton's Laws of Motion

Brief Review

Gravity

Brief Review

Orbits in the Solar System

Today, Newton’s work enables us to calculate and predict the orbits of the planets with marvelous precision. We know eight planets, beginning with Mercury closest to the Sun and extending outward to Neptune. The average orbital data for the planets are summarized in Table 3.2. (Ceres is the largest of the asteroids, now considered a dwarf planet.)

According to Kepler’s laws, Mercury must have the shortest orbital period (88 Earth-days); thus, it has the highest orbital speed, averaging 48 kilometers per second. At the opposite extreme, Neptune has a period of 165 years and an average orbital speed of just 5 kilometers per second.

All the planets have orbits of rather low eccentricity. The most eccentric orbit is that of Mercury (0.21); the rest have eccentricities smaller than 0.1. It is fortunate that among the rest, Mars has an eccentricity greater than that of many of the other planets. Otherwise the pre-telescopic observations of Brahe would not have been sufficient for Kepler to deduce that its orbit had the shape of an ellipse rather than a circle.

The planetary orbits are also confined close to a common plane, which is near the plane of Earth’s orbit (called the ecliptic). The strange orbit of the dwarf planet Pluto is inclined about 17° to the ecliptic, and that of the dwarf planet Eris (orbiting even farther away from the Sun than Pluto) by 44°, but all the major planets lie within 10° of the common plane of the solar system.

In addition to the eight planets, there are many smaller objects in the solar system. Some of these are moons (natural satellites) that orbit all the planets except Mercury and Venus. In addition, there are two classes of smaller objects in heliocentric orbits: asteroids and comets. Both asteroids and comets are believed to be small chunks of material left over from the formation process of the solar system.

In general, asteroids have orbits with smaller semi-major axes than do comets (\(\PageIndex{8}\)). The majority of them lie between 2.2 and 3.3 AU, in the region known as the asteroid belt (see Comets and Asteroids: Debris of the Solar System). As you can see in Table \(\PageIndex{8}\), the asteroid belt (represented by its largest member, Ceres) is in the middle of a gap between the orbits of Mars and Jupiter. It is because these two planets are so far apart that stable orbits of small bodies can exist in the region between them.

Figure \(\PageIndex{8}\): Solar System Orbits. We see the orbits of typical comets and asteroids compared with those of the planets Mercury, Venus, Earth, Mars, and Jupiter (black circles). Shown in red are three comets: Halley, Kopff, and Encke. In blue are the four largest asteroids: Ceres, Pallas, Vesta, and Hygeia (Openstax Astronomy 2nd ed.)

Table \(\PageIndex{8}\): Orbital Data for the Planets

Planet	Semimajor Axis (AU)	Period (y)	Eccentricity	Average Orbital Speed (km/s)	Inclination (°)
Mercury	0.39	0.24	0.21	47.9	7.0
Venus	0.72	0.6	0.01	35.0	3.4
Earth	1	1.00	0.02	29.8	0.0
Mars	1.52	1.88	0.09	24.1	1.9
(Ceres)	2.77	4.6	0.08	17.9	10.6
Jupiter	5.20	11.86	0.05	13.1	1.3
Saturn	9.54	29.46	0.06	9.6	2.5
Uranus	19.19	84.01	0.05	6.8	0.8
Neptune	30.06	164.82	0.01	5.4	1.8

Comets generally have orbits of larger size and greater eccentricity than those of the asteroids. Typically, the eccentricity of their orbits is 0.8 or higher. According to Kepler’s second law, therefore, they spend most of their time far from the Sun, moving very slowly. As they approach perihelion, the comets speed up and whip through the inner parts of their orbits more rapidly.

Rotation

Spin Axis

There are several basic motions of bodies. Rotation is the spinning of an object around its axis. We can put an imaginary stick through Earth’s North and South Poles, representing our planet’s axis. Earth rotates once approximately every 24 hours, while Mars rotates once approximately every 24 hours ans 37 minutes.

At any moment, half of the Earth is illuminated by the Sun, because of the fact that most of our light comes from the Sun. As the Earth rotates from west to east, your location moves from the dark (night) hemisphere into the illuminated (day) hemisphere and back again. The diurnal (daily) motion of the stars, the Sun, and the Moon is a consequence of Earth’s rotation.

Angular Momentum

A concept that is a bit more complex, but important for understanding many astronomical objects, is angular momentum, which is a measure of the rotation of a body as it revolves around some fixed point (an example is a planet orbiting the Sun). The angular momentum of an object is defined as the product of its mass, its velocity, and its distance from the fixed point around which it revolves.

If these three quantities remain constant—that is, if the motion of a particular object takes place at a constant velocity at a fixed distance from the spin center—then the angular momentum is also a constant. Kepler’s second law is a consequence of the conservation of angular momentum. As a planet approaches the Sun on its elliptical orbit and the distance to the spin center decreases, the planet speeds up to conserve the angular momentum. Similarly, when the planet is farther from the Sun, it moves more slowly.

The conservation of angular momentum is illustrated by figure skaters, who bring their arms and legs in to spin more rapidly, and extend their arms and legs to slow down. You can duplicate this yourself on a well-oiled swivel stool by starting yourself spinning slowly with your arms extended and then pulling your arms in. Another example of the conservation of angular momentum is a shrinking cloud of dust or a star collapsing on itself (both are situations that you will learn about as you read on). As material moves to a lesser distance from the spin center, the speed of the material increases to conserve angular momentum.

Further Exploration

• These resources may not be accessible to assistive technologies:
  • The Kepler's Second Law demonstrator from CCNY's ScienceSims project demonstrates how an orbiting planet sweeps out the same area in the same time.
  • Try this simulation of gravity and orbits that lets you move the Sun, Earth, Moon, and space station to see the effects of changing their distances on their gravitational forces and orbital paths. You can even turn off gravity and see what happens.