Kepler's Laws
- According to Newton's Law of Gravity, the acceleration due to gravity at the earth's surface is $\displaystyle g = \frac{GM}{r^2}$ where $M$ is the mass of the earth and $r$ is the radius of the earth. Using $g \approx 9.81 m/sec^2$ and $r \approx 6.37 \times 10^6 m$, approximate the mass of the earth.
- We are unable to stand on the surface of the sun to measure the acceleration of its gravity. But we can find the mass using Kepler's third law, that $\displaystyle \frac{T^2}{a^3} = \frac{4\pi^2}{GM}$, where $T$ is the orbital period, $a$ is the semi-major axis, and $M$ is the mass of the sun. For earth's orbit, $T \approx 3.156 \times 10^7 sec$ and $a \approx 1.496 \times 10^{11} m$. What is the approximate mass of the sun?
- Europa orbits Jupiter with semi-major axis about $6.709 \times 10^8 m$ and period $3.068 \times 10^5 sec$. What is the approximate mass of Jupiter?
- Phobos orbits Mars with semi-major axis about $9.376 \times 10^6 m$ and period $2.755 \times 10^4 sec$. What is the approximate mass of Mars?
- The European Space Agency landed the Philae probe on Comet 67P/Churyumov-Gerasimenko (67P/C-G). Comet 67P/C-G is a short period comet with an orbital period of approximately $2.03 \times 10^8 sec$. It has a perihelion of approximately $1.86 \times 10^{11} m$. What is its aphelion?
- One group of short period comets typically have orbits that take them from aphelion at about the distance of Jupiter from the Sun to the inner solar system for perihelion. This gives them a typical semi-major axis of around $10^{11} m$ to $5 \times 10^{11} m$. What would the periods be for such comets?
- Observing that the fossil record shows major extinction events tend to happen about every 25 million years, it has been theorized that perhaps there is a red or brown dwarf star orbiting the sun, called Nemesis, that has an orbit that disturbs trans-Neptunian bodies every 25 million years or so, leading to much higher chances of cometary impacts on the inner planets. What would the semi-major axis be for an orbit with a period of 25 million years (remember that $\mu$ is given above in units of seconds, not years)? Note: astronomical surveys have looked for such an object and not found anything, so the Nemesis hypothesis is currently considered incorrect.
- Speed is the magnitude of the velocity vector. Show that at perihelion and aphelion, speed is $c/r$ where $c$ is the angular velocity. Hint: there is something special about perihelion and aphelion that will simplify the calculations.
- Derive the following formula for $c$ in terms of perihelion = $r_p$, aphelion = $r_a$, and the standard gravitational parameter $\mu=GM$. $$c = \sqrt{\mu\frac{2r_p r_a}{r_p+r_a}} $$
- Combining the results from the previous 2 problems, show that velocity at perihelion is $$ v_p = \sqrt{\mu\frac{2 r_a}{r_p(r_p+r_a)}} $$ and velocity at aphelion is $$ v_a = \sqrt{\mu\frac{2 r_p}{r_a(r_p+r_a)}} $$
- Halley's comet has a period of about $2.38 \times 10^9 sec$, a perihelion of about $8.766 \times 10^{10} m$, and aphelion of about $5.25 \times 10^{12} m$ (you could deduce this last value from the first two of course). What is the speed of Halley's comet at perihelion?
- Halley's comet has a period of about $2.38 \times 10^9 sec$, a perihelion of about $8.766 \times 10^{10} m$, and aphelion of about $5.25 \times 10^{12} m$ (you could deduce this last value from the first two of course). What is the speed of Halley's comet at aphelion?
- Suppose a comet is observed with a perihelion of $1.30 \times 10^{11} m$ and moving with a speed of $4.50 \times 10^4 m/sec$ at perihelion. What is the period of the orbit of the comet?
- Suppose a comet is observed with a perihelion of $7.50 \times 10^{10} m$ and moving with a speed of $5.00 \times 10^4 m/sec$ at perihelion. What is the period of the orbit of the comet?
- Suppose a comet has a perihelion of $5.50 \times 10^{10} m.$ What speed would the comet need to have at perihelion to be on a "parabolic" orbit and escape the solar system? (Hint: on a parabolic orbit, aphelion=$\infty$.)
- Suppose a comet has a perihelion of $8.00 \times 10^{10} m.$ What speed would the comet need to have at perihelion to be on a "parabolic" orbit and escape the solar system? (Hint: on a parabolic orbit, aphelion=$\infty$.)
- We wish to send a mission to Mars. The minimum fuel transfer orbit is the Hohmann transfer orbit which starts at perihelion at Earth, $1.50 \times 10^{11} m$, and has aphelion at Mars, $2.21 \times 10^{11} m$. What is the initial velocity needed to achieve this orbit? Note: it isn't as unreasonable as it may seem to reach this velocity since we are launching from Earth which is already moving at about $30,000 m/sec$.
- We wish to send a mission to Jupiter. the minimum fuel transfer orbit is the Hohmann transfer orbit which starts at perihelion at Earth, $1.50 \times 10^{11} m$, and has aphelion at Jupiter, $7.78 \times 10^{11} m$. What is the initial velocity needed to achieve this orbit? Note: it isn't as unreasonable as it may seem to reach this velocity since we are launching from Earth which is already moving at about $30,000 m/sec$.
- Suppose Bennett's Law of Gravity says that the gravitational force between two masses is given by $\displaystyle -\frac{GMm}{r}$ (where the denominator is $r$ instead of $r^2$). Show the Kepler's second law still holds for this model.
- With Bennett's Law of Gravity from the previous problem, the key equation in "Writing an Equation for the Orbit" becomes $$ r'' - r(\theta')^2 = - \frac{GM}{r}. $$ Show that if you start from here and make the substitution $u = 1/r$ and write the equation as a differential equation for $u$ as a function of $\theta$, you get a non-linear equation.
- More generally, show that the law of gravitational force $\displaystyle -\frac{GM}{r^n}$ leads to a linear equation for $u$ as a function of $\theta$ only when $n=2$ or $n=3$.
- From Kepler's first law, we know that if we have an equation for $u = 1/r$ as a function of $\theta$, then the solution must have the form $u = B - A\cos(\theta)$, since that is the general form for an ellipse with a focus at the origin and aphelion at $\theta = 0$. Show that this is a solution for the law of gravitational force $\displaystyle -\frac{GM}{r^n}$ only when $n=2$.
- By definition, an ellipse consists of all points with the property that the sum of the distances from the point to two foci is a fixed quantity. Pick the coordinate system so the foci are $(f,0)$ and $(-f,0)$ and the sum of the distances be $2a$, and let $b^2 = a^2 - f^2$. Show the equation of the ellipse is $$ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1$$
- Show that the polar equation $\displaystyle r = \frac{ep}{1-e\cos(\theta)}$ becomes $$\left(\frac{x-f}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1 $$ with $\displaystyle f = \frac{e^2p}{1-e^2}$, $\displaystyle a = \frac{ep}{1-e^2}$, and $\displaystyle b = \frac{ep}{\sqrt{1-e^2}}$.
- Show that $b^2 + f^2 = a^2$ for the values of $a$, $b$, and $f$ in problem 20. Note: problems 23-25 show that $$ r = \frac{ep}{1-e\cos(\theta)}$$ is the equation of an ellipse with a focus at the origin, since you now have the equation in problem 24 is that of the ellipse from problem 23, only translated by $f$, the distance from the focus to the origin.
You may use $G \approx 6.67 \times 10^{-11} m^3 kg^{-1} sec^{-2}$ for the gravitational constant in the following problems.
For the next problems you may find it easier to use the standard gravitational parameter for the sun, $\mu = GM \approx 1.327 \times 10^{20} m^3/sec^2$ rather than having to multiply $G$ and $M$ each time.
You may find it interesting to compare your answers to problems 11 and 12 to the mean speed of the Earth in orbit, which is about $30,000 m/sec$. Note that the Earth's orbit is close to circular (eccentricity 0.0167) so the speed only varies a few percent over the year.
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©1994-2025 Andrew G. Bennett