Journal of Online Mathematics and Its Applications

Volume 6. August 2006. Article ID 1275

Which Planet is Largest?

Phillip Brown and James Braselton

Contents

  1. Introduction
  2. Circular Orbits
  3. Elliptical Orbits
  4. Conclusions
  5. References and Resources

1. Introduction

Jupiter, of course! Well, perhaps not--in this article we interpret the question in a new way: which planet occupies the largest space as it revolves about the Sun? As an extreme case, the volume occupied by Mercury in its orbit about the Sun is surely less than the volume occupied by Earth in its orbit about the Sun. After all, Mercury is much smaller than Earth and much closer to the Sun. More thoughtfully, we pondered: "Pluto is much smaller than Earth but much farther away from the Sun. Which planetís orbit about the Sun occupies a greater volume?" More generally, how do the size of a planet and its distance to the Sun affect the orbit volume of the planet? Finally, a planet's distance to the Sun is not constant. What affect does this have?

The graphic below, from NASA, shows the solar system with the planets in their correct relative sizes. It would be nice to show correct relative distances as well, but that's essentially impossible given the enormous variation in the distances and the limits of a computer screen.

Image: SolarSystem.jpg

Kepler's laws describe planetary motion. You can visit the excellent Kepler Mission site for more information about Johannes Kepler and his laws. However, for this article, we just need his first law:

Kepler's first law
Planets move in elliptical orbits with the Sun at one focus.

Table 1 below, from NASA, gives data on the elliptical orbits of the planets.

Table 1. Planetary Data
Planet Semimajor Axis of Orbit (km) Mean Radius (km) Orbit Eccentricity
Mercury 57,909,175 2,439.7 0.20563
Venus 108,208,930 6,051.8 0.0068
Earth 149,597,890 6,378.14 0.016710
Mars 227,936,640 3,397 0.0934
Jupiter 778,412,020 71,492 0.04839
Saturn 1,426,725,400 60,268 0.0541506
Uranus 2,870,972,200 25,559 0.047168
Neptune 4,498,252,900 24,764 0.00859
Pluto 5,906,380,000 1,151 0.2488

2. Circular Orbits

Because the eccentricities of the planets are small, their orbits are often assumed to be circular. The following image shows the orbits of some of the inner planets.

Image: InnerOrbits.jpg

The animation below shows two planets with circular orbits. One is larger but the other is farther from the Sun. Which planet would generate the greater volume? Run the animation and make a guess.

<img src="CircleA.png"/>

Assuming a circular orbit, a planet's orbit space is the solid obtained by revolving a circular region about the y-axis, which is a torus. Run the animation below to view the process.

<img src="CircleB.png"/>

So, how do we find the volume? In calculus (see any standard text, such as Stewart), we learn the methods of washers and cylindrical shells to find the volume of a solid obtained by revolving a given region about a given line. For Image: Eqn0.gif, we can use either method to find the volume of the solid obtained by revolving the region bounded by the circle Image: Eqn1.gif about the y-axis.

Using the method of cylindrical shells, the volume is:

(1) Image: Eqn2.gif

where we have used that Image: Eqn3.gif because Image: Eqn4.gif is an odd function and Image: Eqn5.gif because the graph of Image: Eqn6.gif is the upper half of the graph of the circle with center at (0, 0) and radius r, Image: Eqn7.gif. (Of course, you could also use the substitutions Image: Eqn8.gif or Image: Eqn9.gif to evaluate Image: Eqn10.gif.)

Let's check our answer using the method of washers:

Image: Eqn11.gif

Good, they agree! Of course, we also recognize our answer as the volume of the circular cylinder of radius r and length Image: Eqn27.gif; we could obtain this cylinder by cutting the torus and straightening it out:

Image: Cylinder.png

We can now use Equation 1 and the data in Table 1 to approximate the orbit volume of each planet. The x0 value is obtained from the second column of Table 1 and the r value from the third column. The results (in km3) are given in the second column of Table 2.

Table 2. Orbit volumes of the planets
Planet Orbit Volume (km3), circular orbit Orbit Volume (km3), elliptical orbit Percent Change
Mercury 6.803775 × 1015 6.731273 × 1015 1.07709%
Venus 7.822796 × 1016 7.822705 × 1016 0.00116%
Earth 1.201277 × 1017 1.201194 × 1017 0.00698%
Mars 5.192004 × 1016 5.180662 × 1016 0.21189%
Jupiter 7.853336 × 1019 7.848736 × 1019 0.05861%
Saturn 1.022925 × 1020 1.022174 × 1020 0.07347%
Uranus 3.702085 × 1019 3.700025 × 1019 0.05567%
Neptune 5.445217 × 1019 5.445117 × 1019 0.00184%
Pluto 1.544549 × 1017 1.520362 × 1017 1.59087%

3. Elliptical Orbits

Of course, the orbits of the planets are not perfect circles. The image below shows the orbits of some of the outer planets.

Image: OuterOrbits.jpg

In this section we will attempt to improve our approximations by finding the volume of the region occupied by a sphere in an elliptical orbit. First, run the animation below. The orbit of the smaller planet is approximately circular while the orbit of the larger planet is elliptical. Which would generate the larger volume?

<img src="EllipseA.png"/>

Run the animation below to see the solid generated by a planet in an elliptical orbit.

<img src="EllipseB.png"/>

Now let's do the math. Consider an elliptical shell with height h and base in the x-z plane given by Image: Eqn12.gif. Using the arc length formula in rectangular coordinates, the circumference of the ellipse is

Image: Eqn13.gif

Hence the surface area of the shell can be computed by multiplying the circumference of the ellipse by the height:

(2) Image: Eqn14.gif

Now assume that the major axis is the line segment connecting Image: Eqn15.gif and Image: Eqn16.gif and that the ellipse has eccentricity e, which is defined by

Image: Eqn17.gif

Then Image: Eqn18.gif and rewriting Equation 2 in terms of the length of the major axis and the eccentricity of the ellipse gives us the elliptical shell surface area:

(3) Image: Eqn19.gif

where E(z) is the complete elliptic integral of the second kind:

(4) Image: Eqn20.gif

You can learn more about elliptic integrals at the MathWorld site. Now, to find the volume of the region occupied by the sphere

Image: Eqn21.gif

as it travels about its elliptical path,

Image: Eqn22.gif

in the x-z plane (y = 0), we use the method of elliptical shells and evaluate the volume of the elliptical torus:

(5) Image: Eqn23.gif

From the Maclaurin series expansion for E (Equation 4), observe that for Image: Eqn24.gif, Image: Eqn25.gif and Equation 5 agrees with Equation 1:

Image: Eqn26.gif

We now use Equation 5, the planetary data in Table 1, and a computer algebra system to find the volume of the elliptical torus for each planet. The x0 value is obtained from the second column of Table 1, the r value from the third column, and the e value from the fourth column. The approximate orbit volumes (in km3) of the planets are given in the third column of Table 2.

The last column of Table 2 shows the percent by which the volume of the circular orbit exceeds the volume of the elliptical orbit. With the exception of Mercury, Pluto, and (possibly) Mars, we see that the error is quite small, which helps us understand why the assumption of circular orbits is often made.

4. Conclusions

We have used two applications of integration to compute the orbit volumes of the planets. We used the methods of cylindrical shells and washers, assuming circular orbits, and the method of elliptical shells, assuming elliptical orbits. The largest planet, in terms of orbit volume, turns out to be Saturn, not Jupiter! The orbit volumes of Earth and Pluto are very close; Pluto has the larger volume because its enormous distance from the Sun outweighs its much smaller size. The following table gives a complete ranking of the planets (from smallest to largest) in terms of ordinary volume and orbit volume:

Table 3. Planet rankings
Planet Volume Rank Orbit Volume Rank
Mercury 2 1
Venus 4 3
Earth 5 4
Mars 3 2
Jupiter 9 8
Saturn 8 9
Uranus 7 6
Neptune 6 7
Pluto 1 5

Our colleague Kevin Phillips pointed out that the volume of the Sun is about 1.41 × 1018 km3, larger than the orbit volume of five of the nine planets! We should also note that, as we write this, the definition of planet is being reconsidered. There may soon be many additional planets whose orbit volumes can be computed.

5. References and Resources

Source Files

Mathematica was used to performs many of the calculations illustrated here as well as to generate most of the graphics and all of the animations. You may download the Mathematica code.