Exercises: The laws of planetary motion

Johannes Kepler deduced three laws describing the motion of planets around the Sun from the observations of Tycho Brahe.

1. a) State Kepler's first law (law of orbits).
2. b) Draw an elliptical orbit in your notebook and indicate on it the two foci of the ellipse, the position of the Sun, perihelion (the point closest to the Sun) and aphelion (the point farthest away).
3. c) State what the semi-major axis $a$ of the orbit is, and write its expression in terms of the distances at perihelion, $r_p$, and aphelion, $r_a$.

You can do it on paper and send a photo with your answer.

1. a) State Kepler's second law (law of areas).
2. b) A planet moves along its elliptical orbit, passing through perihelion and aphelion. Apply the law of areas to reason in which of these two points it moves faster.
3. c) The areal velocity (area swept out by the radius vector per unit time) is constant because the planet's angular momentum is conserved. Explain qualitatively why a shorter radius vector at perihelion implies a higher velocity.

1. a) State Kepler's third law (law of periods) and write its mathematical expression, $\frac{T^2}{a^3} = C$, stating the meaning of $T$ and $a$.
2. b) The constant is equal to $C = \frac{4\times\pi^2}{G\times M}$, where $M$ is the mass of the central body. Explain whether $C$ is the same for all planets in the Solar System and whether it would also be the same for Jupiter's moons.

The Earth passes through perihelion at a distance of $r_p = 1.471\times10^{11}\ \text{m}$ from the Sun and through aphelion at $r_a = 1.521\times10^{11}\ \text{m}$. The figure shows the orbit with both distances (not to scale):

1. a) Calculate the semi-major axis of the Earth's orbit, $a = \frac{r_p + r_a}{2}$.
2. b) Given that $1\ \text{UA} = 1.496\times10^{11}\ \text{m}$ (astronomical unit: mean Earth–Sun distance), express the result in astronomical units and comment on what you observe.

For bodies orbiting the Sun, if the period $T$ is expressed in years and the semi-major axis $a$ in astronomical units, Kepler's third law reduces to $T^2 = a^3$. Mercury orbits with a semi-major axis $a = 0.387\ \text{UA}$.

1. a) Calculate Mercury's orbital period, in years.
2. b) Express it in Earth days ($1$ year $= 365.25$ days).

An asteroid in the main belt takes $T = 8$ years to complete its orbit around the Sun. For bodies orbiting the Sun, $T^2 = a^3$ holds, with $T$ in years and $a$ in astronomical units.

1. a) Rearrange the above expression to solve for the semi-major axis $a$.
2. b) Calculate its value, in $\text{UA}$.

A student claims: "as $\frac{T^2}{a^3} = C$ is a universal constant, its value is the same for the Earth around the Sun as for the Moon around the Earth".

1. a) Explain why this statement is false, taking into account that $C = \frac{4\times\pi^2}{G\times M}$, where $M$ is the mass of the central body.
2. b) State a set of bodies for which the constant would coincide with that of Jupiter's moons.

Draw an elongated elliptical orbit in your notebook with the Sun located at one of its foci.

1. a) Mark two equal time intervals, one centered at perihelion and the other at aphelion, and shade the areas swept out by the radius vector in both sections.
2. b) According to the law of areas, what relationship exists between the two shaded areas?
3. c) Reason, from the diagram, in which section the planet covers a longer arc in the same time, and what this implies about its velocity.

You can do it on paper and send a photo with your answer.

Mars orbits the Sun with a semi-major axis $a = 1.524\ \text{UA}$. For bodies orbiting the Sun, $T^2 = a^3$ with $T$ in years and $a$ in astronomical units.

1. a) Calculate the length of the Martian year, in Earth years.
2. b) Express the result in days ($1$ year $= 365.25$ days).

Venus completes its orbit around the Sun in $T = 0.615$ years. For bodies orbiting the Sun, $T^2 = a^3$ with $T$ in years and $a$ in astronomical units; $1\ \text{UA} = 1.496\times10^{11}\ \text{m}$.

1. a) Calculate the semi-major axis of Venus's orbit, in $\text{UA}$.
2. b) Express it in kilometers, using scientific notation.

A team of astronomers discovers an object whose orbital period is $T = 27$ years. For bodies orbiting the Sun, $T^2 = a^3$ with $T$ in years and $a$ in astronomical units.

1. a) Calculate the semi-major axis of its orbit, in $\text{UA}$.
2. b) Compare it with Neptune's orbit ($a \approx 30\ \text{UA}$): does the object orbit closer to or farther from the Sun than Neptune?

Kepler verified his third law with data from several planets. Earth has $a = 1\ \text{UA}$ and $T = 1$ year; Jupiter has $a = 5.20\ \text{UA}$ and $T = 11.86$ years.

1. a) Calculate the ratio $\frac{T^2}{a^3}$ for Earth, in $\text{years}^2/\text{AU}^3$.
2. b) Calculate the same ratio for Jupiter and check that it matches the previous one (allowing for rounding).
3. c) Explain why this ratio is only identical for bodies orbiting the same central body.

The eccentricity of an ellipse is $e = \frac{c}{a}$, where $a$ is the semi-major axis and $c$ is the distance from the center of the ellipse to one of the foci. For the Earth: $r_p = 1.471\times10^{11}\ \text{m}$ (perihelion) and $r_a = 1.521\times10^{11}\ \text{m}$ (aphelion). Observe the geometry of the orbit in the figure (not to scale):

1. a) Calculate the semi-major axis, $a = \frac{r_p + r_a}{2}$.
2. b) Calculate the focal distance, $c = \frac{r_a - r_p}{2}$, and the eccentricity $e$ of the Earth's orbit.
3. c) A value of $e = 0$ corresponds to a circle and a value close to $1$ to a highly elongated ellipse. Comment on the shape of the Earth's orbit according to the value obtained.

Ceres, the largest body in the asteroid belt, has an orbital period of $T = 4.60$ years. For bodies orbiting the Sun, $T^2 = a^3$ with $T$ in years and $a$ in astronomical units; $1\ \text{UA} = 1.496\times10^{11}\ \text{m}$.

1. a) Calculate the semi-major axis of Ceres's orbit, in $\text{UA}$.
2. b) Express its mean distance to the Sun in meters, using scientific notation.

A comet has a semi-major axis twice that of another comet in the same system.

1. a) Show, using $T^2 = a^3$, that if $a' = 2\times a$ then $T' = \sqrt{8}\times T \approx 2.83\times T$.
2. b) An asteroid orbits the Sun with $a = 2\ \text{UA}$. Calculate its period, in years.
3. c) Reason, without carrying out new calculations, whether a planet located at half the distance from the Sun as the Earth would have a year longer or shorter than Earth's.