1.3: Modern Astronomy

The Science of Astronomy

From here we will shift to tracking the story of astronomy after the end of the middle ages in Europe. Before starting we want to acknowledge the many contributions to astronomy that people have made all around the world. Again, listing all contributions would take yet another textbook on top of the one for ancient astronomy. Always remember that astronomy, and science in general, is a collaborative effort. For every person we highlight there were dozens more that helped them collect data, make calculations, and brainstorm solutions to problems.

The Heliocentric Model

By the 1500s, the old Ptolemaic system was no longer working well. It needed significant adjustments to predict the positions of the planets correctly. To be able to accurately predict the positions of planets, Nicolaus Copernicus, a Polish astronomer and mathematician, revived the heliocentric model of Aristarchus. Figure \(\PageIndex{1}\) is a diagram of the Solar System by Copernicus, with the Sun at the center. He concluded that Earth is a planet and that all the planets orbit the Sun. Using this concept, he was able to correctly place the planets, in the correct order: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Copernicus described his ideas in his book De Revolutionibus Orbium Coelestium (On the Revolution of Celestial Orbs), published in 1543, the year of his death.

Figure \(\PageIndex{1}\) : The Heliocentric Model. Copernicus placed the Sun, labeled Sol, at the center, with the planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn—orbiting outward, replacing Ptolemy's Earth-centered model. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{1}\).

Copernicus argued that the apparent motion of the Sun about Earth during the course of a year could be represented by motion of Earth about the Sun. In the geocentric model, all of the stars were on a solid shell, called the celestial sphere, that rotated around the Earth. Copernicus reasoned that the apparent rotation of the celestial sphere could be explained by assuming that Earth rotates while the celestial sphere is stationary. He also discovered that the planets closest to the Sun have faster orbits than those farther away from the Sun.

With his theory, he was able to explain the retrograde motions of the planets without epicycles. Figure \(\PageIndex{2}\) demonstrates the motion of Earth and an outer planet, let's call it Mars, in a heliocentric system. Earth travels around the Sun in the same direction as Mars. Because Earth's orbital speed is faster, it overtakes Mars periodically, like a faster race car on the inside track. The path of the planet among the stars is on the right side of the figure. Normally, planets move eastward in the sky over the weeks and months as they orbit the Sun, but from positions B to D, Mars appears to drift backward, moving west in the sky. Even though Mars is actually moving to the east, the faster-moving Earth has overtaken it, and as Earth passes, it looks like Mars is moving backward. As Earth rounds its orbit toward position E, Mars again takes up its apparent eastward motion in the sky.

Figure \(\PageIndex{2}\) : Retrograde Motion. Because Earth moves faster than Mars, it periodically overtakes the planet, making Mars appear to briefly reverse direction, or move retrograde, against the background stars. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{2}\).

His ideas, although not widely accepted until more than a century after his death, were much discussed among scholars. One of the objections raised against the heliocentric theory was that if Earth were moving, we would all sense or feel this motion. Solid objects would be ripped from the surface, and a ball dropped from a great height would not strike the ground directly below it. In Copernicus' time, in fact, few people thought there were ways to prove whether the heliocentric or the older geocentric system was correct. At the time, science valued human thought over observations and data. This meant that instead of conducting experiments or collecting data to test the heliocentric model, it was debated as a purely philosophical concept.

Regardless of the debate, the heliocentric model already passed the test of accurately predicting the positions of planets. Copernicus predicted that, if Venus circles the Sun, the planet should go through the full range of phases just as the Moon does. Figure \(\PageIndex{3}\) demonstrates the phases of Venus as seen from Earth. Also, there would be a period of time that we can't see Venus because it is on the other side of the Sun from the Earth. Before the telescope, no one tested these predictions.

Figure \(\PageIndex{3}\) : Phases of Venus. As Venus orbits the Sun, sunlight illuminates different portions of the side facing Earth, producing gibbous, half, and crescent phases just as the Moon does. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{3}\).

The Beginning of Modern Science

In the late 1500s of Italy, Galileo Galilei began applying the modern scientific concepts of observation, experimentation, and the testing of hypotheses through careful quantitative measurements. His major contributions were in the field of mechanics, the study of motion and the actions of forces on bodies. He argued that a force is required not only to start an object moving from rest but also to slow down, stop, speed up, or change the direction of a moving object. He also studied the way objects accelerate, or change their speed or direction of motion. Galileo watched objects as they fell or rolled down a ramp. He found that such objects accelerate uniformly. In equal intervals of time they gain equal increments in speed. Galileo formulated these newly found laws in precise mathematical terms that enabled future experimenters to predict how far and how fast objects would move in various lengths of time.

Galileo's Astronomical Observations

After acquiring a telescope, known then as a spyglass, Galileo decided to use it to observe objects in the sky. Beginning his astronomical work late in 1609, Galileo found that many stars too faint to be seen with the unaided eye became visible with his telescope. In particular, he found that some nebulous blurs resolved into many stars, and that the Milky Way, the strip of whiteness across the night sky, was also made up of a multitude of individual stars.

Examining the planets, Galileo found four moons orbiting Jupiter in times ranging from just under 2 days to about 17 days. This discovery was particularly important because it showed that not everything has to revolve around Earth. Furthermore, it demonstrated that there could be centers of motion that are themselves in motion. Defenders of the geocentric view had argued that if Earth was in motion, then the Moon would be left behind because it could hardly keep up with a rapidly moving planet. Yet, here were Jupiter's moons doing exactly that.

Galileo proceeded to observe the phases of Venus in Figure \(\PageIndex{3}\), demonstrating that it must orbit around the Sun, so that we see different parts of its daylight side at different times. These observations could not be reconciled with the geocentric model, in which Venus orbited around Earth. Galileo also observed the Moon and saw craters, mountain ranges, valleys, and flat, dark areas that he thought might be water. These discoveries showed that the Moon might be not so dissimilar to Earth, suggesting that Earth, too, could belong to the realm of celestial bodies.

Changing the Universe

Galileo realized that his observations agreed with the heliocentric model of the Solar System. In Roman Catholic Italy, this was not a popular philosophy, for Church authorities still upheld the ideas of Aristotle and Ptolemy, and they had powerful political and economic reasons for insisting that Earth was the center of creation. Galileo challenged the geocentric model using his observations of nature. It was primarily because of Galileo that, in 1616, the Church issued a prohibition decree stating that the Copernican doctrine was false and absurd and not to be held or defended.

After Galileo's work, it became increasingly difficult to deny the heliocentric model, and Earth was slowly moved from its central position in the universe to one of the planets orbiting the Sun. Initially, however, Galileo met with a great deal of opposition. The Roman Catholic Church had Galileo appear before the Inquisition to answer charges that his work was heretical. He was ultimately condemned to house arrest. His books were on the Church's forbidden list until 1836, although in many countries they were widely read and discussed. Not until 1992 did the Catholic Church admit publicly that it had erred in the matter of censoring Galileo's ideas.

Shifting from the geocentric to the heliocentric model, not only moved Earth from the center of the universe, but it also established observation, experimentation, and testing of hypotheses as the standard for scientific research. What followed was a rapid expansion of astronomical knowledge.

Planetary Motion

At about the time that Galileo was beginning his experiments with falling bodies, the observer Tycho Brahe and the mathematician Johannes Kepler were observing and analyzing the motions of the planets. Brahe, a Danish astronomer, gained the patronage of the Danish King Frederick II. At the age of 30, Brahe was able to establish an astronomical observatory on the North Sea island of Hven. Brahe made 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. These data were extremely valuable, but Brahe didn't have the ability to analyze them and develop a better model.

Near the end of his life, Brahe moved to Prague, where he met a young mathematician, Johannes Kepler, to assist him in analyzing his extensive planetary data. Through his analysis of the motions of the planets, 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.

Kepler's Laws

The path of an object through space is called its orbit. Kepler initially assumed that the orbits of planets were circles, but circular orbits did not match the 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. Figure \(\PageIndex{4}\) includes the conics ections, a family of curves derived from a cone. Next to the circle, the ellipse is the simplest kind of conic.

Figure \(\PageIndex{4}\) : Conic Sections. A circle, ellipse, parabola, and hyperbola are each produced by slicing a cone at a different angle, which is why these curves are called conic sections. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{4}\).

In a circle, the center is a special point. 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 points inside the ellipse are called its foci, singular: focus, a word invented for this purpose by Kepler. Figure \(\PageIndex{5}\) demonstrates how to draw an ellipse. We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack. If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is the length of the string. The tacks are at the two foci of the ellipse.

Figure \(\PageIndex{5}\) : Drawing an Ellipse. Looping a string around two thumbtacks and tracing it taut with a pencil produces an ellipse, because the string's fixed length keeps the sum of the distances to the two foci constant. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{5}\).

The widest diameter of the ellipse is called its major axis. Half this distance, the distance from the center of the ellipse to one end, is the semimajor axis. The semimajor axis is usually used to specify the size of the ellipse. For example, the semimajor axis of the orbit of Mars, which is also the planet's average distance from the Sun, is 228 million kilometers.

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. If the foci 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. Therfore, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius. Next, we can make ellipses of various extended lengths by varying the spacing of the tacks as long as they are not farther apart than the length of the string. The greater the eccentricity, the more elongated is the ellipse. The maximum eccentricity is 1, where the ellipse becomes flat.

The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. 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, is very similar to a circle. But the difference turned out to be critical for understanding planetary motions. Kepler's first law is that the orbits of all the planets are ellipses. At this point, he discarded the ancient Greek ideal of the perfect circle.

Kepler's second law deals with the speed with which each planet moves along its ellipse, also known as its orbital speed. Kepler discovered that Mars 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 farther from the Sun, positions 1 and 2 in Figure \(\PageIndex{6}\), the elastic line is stretched a lot, and the planet does not move so fast. Closer to the Sun, as in positions 3 and 4, the line is not stretched as much, and the planet moves rapidly. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves, demonstrated by the colored regions in the figure. Kepler found that in equal intervals of time (t), the areas swept out in space by this imaginary line are always equal. In the figure, the area of the region A from 1 to 2 is the same as that of region B 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, in most orbits that speed of a planet orbiting its star, or moon orbiting its planet, tends to vary because the orbit is elliptical.

Figure \(\PageIndex{6}\) : Kepler's Second Law: The Law of Equal Areas. As a planet moves along its elliptical orbit, an imaginary line from the Sun sweeps out equal areas in equal times, so the planet moves faster when closer to the Sun and slower when farther away. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{6}\).

Kepler's next goal was to determine why the orbits of the planets were spaced as they are and to find a mathematical pattern in their movements. 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, or

\[P^2 \propto a^3 \nonumber\]

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 equal. One AU is the average distance between Earth and the Sun and is approximately equal to 1.5 x 10⁸ kilometers. In these units,

\[P^2 = a^3 \nonumber\]

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. 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 P², is \(1.88^2 = 3.53\). According to the equation for Kepler's third law, this equals the cube of its semimajor axis, or a³. So what number must be cubed to give 3.53? The answer is 1.52 since 1.52 x 1.52 x 1.52 = 3.53. Thus, Mars' semimajor axis in astronomical units must be 1.52 AU. In other words, to go around the Sun in a little less than two years, Mars must be about 50% farther from the Sun than Earth is.

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. The short version is equal areas in equal times.
• Kepler's third law: The square of a planet's orbital period is directly proportional to the cube of the semimajor axis of its orbit, P²=a³

Kepler's three laws provide 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: they do not help us understand what forces of nature constrain the planets to follow this particular set of rules.

Gravity

Sir Isaac Newton, an English physicist and mathematician, developed 3 laws of motion:

• Newton's first law: Every object will continue to be in a state of rest or move at a constant speed in a straight line unless it is compelled to change by an outside force.
• Newton's second law: The change of motion of a body is proportional to and in the direction of the force acting on it.
• Newton's third law: For every action there is an equal and opposite reaction or: the mutual actions of two bodies upon each other are always equal and act in opposite directions.

According to Newton's first law, it is the straight line that defines the most natural state of motion. But the planets move in ellipses, not straight lines. There must be some type of force bending their paths. That force, Newton proposed, was gravity. In Newton's time, gravity was something associated with Earth alone. Everyday experience shows us that Earth exerts a gravitational force upon objects at its surface. If you drop something, it accelerates toward Earth as it falls. Newton's insight was that Earth's gravity might extend as far as the Moon and produce the force required to curve the Moon's path from a straight line and keep it in its orbit. He further hypothesized that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies. If so, the attractive force between the Sun and each of the planets could keep them in their orbits.

The precise mathematical description of gravitational force had to dictate that the planets move exactly as Kepler had described them, as expressed in Kepler's three laws. Also, that gravitational force had to predict the correct behavior of falling bodies on Earth, as observed by Galileo. Newton concluded that the magnitude of the force of gravity must decrease with increasing distance between the Sun and a planet in proportion to the inverse square of their separation. In other words, if a planet were twice as far from the Sun, the force would be \(\left(\frac{1}{2}\right)^2\), or \(\frac{1}{4}\) as large. Put the planet three times farther away, and the force is \(\left(\frac{1}{3}\right)^2\), or \(\frac{1}{9}\) as large.

Newton also concluded that the gravitational attraction between two bodies must be proportional to their masses. The more mass an object has, the stronger the pull of its gravitational force. The gravitational attraction between any two objects is therefore given by:

\[F_{\text{gravity}} = G\frac{M_1 M_2}{R^2} \nonumber\]

where F_gravity is the gravitational force between two objects, M₁ and M₂ are the masses of the two objects, and R is their separation. G is a constant number known as the universal gravitational constant.

According to Newton's second law, forces cause acceleration. Newton's universal law of gravitation says that the force acting upon and the acceleration of an object toward Earth should be inversely proportional to the square of its distance from the center of Earth. Objects like apples at the surface of Earth, at a distance of one Earth-radius from the center of Earth, are observed to accelerate downward at 9.8 meters per second per second, 9.8 m/s². It is this force of gravity on the surface of Earth that gives us our sense of weight. Unlike your mass, which would remain the same on any planet or moon, your weight depends on the local force of gravity. So you would weigh less on Mars and the Moon than on Earth, even though there is no change in your mass.

Gravity is a “built-in” property of mass. Whenever there are masses in the universe, they will interact via the force of gravitational attraction. The more mass there is, the greater the force of attraction. Here on Earth, the largest concentration of mass is the planet we stand on, and its pull dominates the gravitational interactions we experience. Newton's law also implies that gravity never becomes zero. It quickly gets weaker with distance, but it continues to act to some degree no matter how far away you get. The pull of the Sun is stronger at Mercury than at Pluto, but it can be felt far beyond Pluto. The Sun's gravitational pull joins with the pull of billions of other stars to create the gravitational pull of our Milky Way Galaxy. That force, in turn, can make other smaller galaxies orbit around the Milky Way.

Why is it then, that the astronauts in space appear to have no gravitational forces acting on them in images like Figure \(\PageIndex{7}\)? The astronauts are only a few hundred kilometers above the surface of Earth, which is not a significant distance compared to the size of Earth. The gravity should not be that much weaker. The astronauts feel weightless because they are falling. An object in orbit around Earth is in a constant state of falling toward Earth and missing, also called free fall.

Figure \(\PageIndex{7}\) : Astronauts in Free Fall. While orbiting Earth, astronauts are in constant free fall, which produces the sensation of weightlessness even though gravity is still acting on them. (CC BY 4.0; Fraknoi, et al. via Openstax Astronomy 2nd ed.) Accessible description of Figure \(\PageIndex{7}\).

Orbital Motion and Mass

Kepler's laws describe the orbits of the objects whose motions are described by Newton's laws of motion and the law of gravity. Knowing that gravity is the force that attracts planets toward the Sun, however, allowed Newton to rethink Kepler's third law. He added the masses of the Sun, M₁, and the planet, M₂, both expressed in units of the Sun's mass. The adapted version of Kepler's third law is:

\[a^3 = (M_1 + M_2) \times P^2 \nonumber\]

where a is the semimajor axis and P is the orbital period.

How did Kepler miss this factor? In units of the Sun's mass, the mass of the Sun is 1, and in units of the Sun's mass, the mass of a typical planet is a negligibly small factor. This means that the sum of the Sun's mass and a planet's mass, \((M_1 + M_2)\), is very, very close to 1. The tiny mass of the planets compared to the Sun is the reason that Kepler did not realize that both masses had to be included in the calculation.

Including the mass term allows us to use this formula in a new way. If we can measure the distances and orbital periods of objects acting under their mutual gravity, then the formula will permit us to calculate their masses. For example, we can calculate the mass of the Sun by using the distances and orbital periods of the planets, or the mass of Jupiter by noting the motions of its moons. Astronomers have made a lot of use out of this tool since it was developed.

Astronomy Now

This review of astronomy history sets the stage for the rest of this textbook. We have reviewed how modern science developed over time and learned the basic tools that astronomers use to learn about celestial objects. The only thing left to discuss is how astronomy functions now, and what differentiates it from other fields of science.

When we reviewed the scientific method, there was a lot of discussion about experiments. This word is often related to a scientist in a white lab coat mixing chemicals in a laboratory. That kind of experimentation is rare in astronomy since it's impossible to put a group of stars into a test tube or to order another comet from a scientific supply company.

As a result, astronomy is sometimes called an observational science. We often make our tests by observing many samples of the kind of object we want to study and noting carefully how different samples vary. New instruments and technology can let us look at astronomical objects from new perspectives and in greater detail. Our hypotheses are then judged in the light of this new information, and they pass or fail in the same way we would evaluate the result of a laboratory experiment.

Much of astronomy is also a historical science. What we observe has already happened in the universe and we can do nothing to change it. In the same way, a geologist cannot alter what has happened to our planet, and a paleontologist cannot bring an ancient animal back to life. While this can make astronomy challenging, it also gives us fascinating opportunities to discover the secrets of our cosmic past.

Examples and Exercises

Example: Calculating Orbital Periods

Imagine an object is traveling around the Sun. What would be the orbital period of the object if its orbit has a semimajor axis of 50 AU?

Solution

From Kepler's third law, we know that (when we use units of years and AU)

\[P^2 = a^3 \nonumber\]

If the object's orbit has a semimajor axis of 50 AU (a = 50), we can cube 50 and then take the square root of the result to get P:

\[P = \sqrt{a^3} = \sqrt{50 \times 50 \times 50} = \sqrt{125{,}000} = 353.6 \text{ years} \nonumber\]

Therefore, the orbital period of the object is about 350 years. This would place our hypothetical object beyond the orbit of Pluto.

Exercise: Orbital Periods

What would be the orbital period of an asteroid (a rocky chunk between Mars and Jupiter) with a semimajor axis of 3 AU?

Answer

\(P = \sqrt{3 \times 3 \times 3} = \sqrt{27} = 5.2 \text{ years}\)

Example: Kepler's Third Law

Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate \(p^2\) and \(a^3\), and verify that they obey Kepler's third law. Venus' orbital period is 0.62 year, and its semimajor axis is 0.72 AU. Earth's orbital period is 1.00 year, and its semimajor axis is 1.00 AU.

Solution

We can use the equation for Kepler's third law, P² ∝ a³. For Venus, \(P^2 = 0.62 \times 0.62 = 0.38\) and \(a^3 = 0.72 \times 0.72 \times 0.72 = .037\). The square of the orbital period (0.38) approximates the cube of the semimajor axis (0.37). Therefore, Venus obeys Kepler's third law. For Earth, \(P^2 = 1.00 \times 1.00 = 1.00\) and \(a^3 = 1.00 \times 1.00 \times 1.00 = 1.00\). The square of the orbital period (1.00) approximates (in this case, equals) the cube of the semimajor axis (1.00). Therefore, Earth obeys Kepler's third law.

Exercise: Kepler's Third Law

Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate P² and a³, and verify that they obey Kepler's third law. Saturn's orbital period is 29.46 years, and its semimajor axis is 9.54 AU. Jupiter's orbital period is 11.86 years, and its semimajor axis is 5.20 AU.

Answer

For Saturn, \(P^2 = 29.46 \times 29.46 = 867.9\) and \(a^3 = 9.54 \times 9.54 \times 9.54 = 868.3\). The square of the orbital period (867.9) approximates the cube of the semimajor axis (868.3). Therefore, Saturn obeys Kepler's third law.

Example: Calculating Weight

By what factor would a person's weight at the surface of Earth change if Earth had its present mass but eight times its present volume?

Solution

With eight times the volume, Earth's radius would double. This means the gravitational force at the surface would reduce by a factor of \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\), so a person would weigh only one-fourth as much.

Exercise: Calculating Weight

By what factor would a person's weight at the surface of Earth change if Earth had its present size but only one-third its present mass?

Answer

With one-third its present mass, the gravitational force at the surface would reduce by a factor of \(\frac{1}{3}\), so a person would weigh only one-third as much.

Example: Measuring Stellar Mass

A planet like Earth is found orbiting its star at a distance of 1 AU in \(0.71\) Earth-year. Can you use Newton's version of Kepler's third law to find the mass of the star? (Remember that compared to the mass of a star, the mass of an earthlike planet can be considered negligible.)

Solution

In the formula \(a^3 = (M_1 + M_2) \times P^2\), the factor \(M_1 + M_2\) would now be approximately equal to \(M_1\) (the mass of the star), since the planet's mass is so small by comparison. Then the formula becomes \(a^3 = M_1 \times P^2\), and we can solve for \(M_1\):

\[M_1 = \frac{a^3}{P^2} \nonumber\]

Since \(a = 1, a^3 = 1\), so

\[M_1 = \frac{1}{P^2} = \frac{1}{0.71^2} = \frac{1}{0.5} = 2 \nonumber\]

So the mass of the star is twice the mass of our Sun. (Remember that this way of expressing the law has units in terms of Earth and the Sun, so masses are expressed in units of the mass of our Sun.)

Exercise: Kepler's Third Law

Suppose a star with twice the mass of our Sun had an earthlike planet that took 4 years to orbit the star. At what distance (semimajor axis) would this planet orbit its star?

Answer

Again, we can neglect the mass of the planet. So \(M_1 = 2\) and \(P = 4\) years. The formula is \(a^3 = M_1 \times P^2\), so \(a^3 = 2 \times 4^2 = 2 \times 16 = 32\). So a is the cube root of 32. To find this, you can use a calculator, and get the answer 3.2 AU.

Further Exploration

Further Exploration

• Watch this animation of the phases of Venus that includes its distance from Earth as it orbits the Sun.
• In 1971, Apollo 15 astronaut David Scott took a hammer and feather to the Moon and demonstrated that all objects fall at the same rate, as Galileo proposed. Watch the video ofthe hammer and feather drop to see how these objects fall without an atmosphere to slow the feather's fall.
• For more information about the life and work of Galileo, visit the Galileo Project at Rice University.
• 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.