Universal Gravitation and Kepler's Laws Notes

Kepler's Laws

• The topic is Kepler's Laws.

Seed Question

• Which planet in our Solar System has the fastest orbital speed? Why?

• A table is presented with data for planets in our solar system:

  • Planet, Period (yr), Average radius (au), $T^2/R^3$ ($yr^2/au^3$)

  • Mercury: 0.241, 0.39, 0.98

  • Venus: 0.615, 0.72, 1.01

  • Earth: 1.00, 1.00, 1.00

  • Mars: 1.88, 1.52, 1.01

  • Jupiter: 11.8, 5.20, 0.99

  • Saturn: 29.5, 9.54, 1.00

  • Uranus: 84.0, 19.18, 1.00

  • Neptune: 165, 30.06, 1.00

  • Pluto: 248, 39.44, 1.00

Exploration

• Orbits WS Kepler's 1st Law Kepler's 2nd Law

Big Idea: Kepler's Laws

• 1st Law: Planets travel in elliptical orbits with the sun at one of the foci.

• 2nd Law: Planets sweep out equal areas in equal time.

• 3rd Law: $T^2/R^3 = 1$ (This constant is 1 only for objects orbiting the sun.)

  • R is the average distance between the planet and the sun.

  • Units for R are in astronomical units (au).

  • 1 au is equal to the distance between the Earth and the sun, which is $1.4957 x 10^{11} m$.

Finding the Mass of the Earth

Seed Question

• How do you think we first determined the mass of the Earth? What would you need to know?

Exploration

• Estimate the mass of Earth using the Moon’s orbit around it.

• Apply the 3-step plan for forces:

  • Step 1: Sketch

  • Step 2: $F<em>{net}$ toward center = $m</em>ac$

  • Step 3: Plug in the values

    • $R = 60R_E = 60(6.4 x 10^6 m) = 3.8 x 10^8 m$ (Ancient Greeks determined the distance from the Earth to the Moon using shadows and geometry)

    • $T = 27 days (24 hr/day)(3600 s/hr) = 2.3 x 10^6 s$ (approximate period of Moon’s orbit around the Earth).

    • Hint: $v = \frac{2\pi R}{T}$ Rewrite equation and sub in values to find $m_E$.

Big Idea

• $m_E = \frac{4\pi^2 r^3}{GT^2}$

• Careful calculation gives: $m_E = 6.0 x 10^{24} kg$

Orbits

Seed Question

• What does it mean to be weightless? What about in orbit around the Earth?

Exploration

• Kepler found that $\frac{T^2}{r^3} = constant$ for all objects orbiting the sun.

• Show why $\frac{T^2}{r^3} = constant$, using Newton’s Universal Law of Gravity, applied to an object in circular orbit around the sun.

• Find Kepler’s constant algebraically first, in terms of $m_{sun}$, G, and any other necessary constants. What is the value of the constant? Units?

Big Idea

• All orbit problems are forces problems.

• Sketch the orbit with $F_{grav}$ on the orbiting object m.

• $F<em>{Net}$ toward center = $m</em>ac$ ($v = \frac{2\pi r}{T}$)

• $\frac{GMm}{r^2} = m\frac{4\pi^2 r}{T^2}$

• Note: orbiting mass m always cancels out.

• Solve for the desired parameter: M, r, or T.

Universal Gravitational Potential Energy

Seed Question

• What are the differences between the two $F_g$ equations?

  • $F_g = mg$ (planet specific)

  • $F<em>g = \frac{Gm</em>1m_2}{R^2}$ (always applicable)

  • g is the acceleration due to gravity or “field of gravity.”

  • The farther the mass moves, the more significant the decrease in g.

Exploration

• (Gravity is constant) What is the area under a $F_{grav}$ vs h graph? (Draw a graph) (an object being lifted up)

• We say the work done by gravity ($PE<em>g$) is the area under a $F</em>{grav}$ vs h graph

• In general: $W = Fd cos(\Theta)$

  • $F=mg$

  • $\Theta = 180$

Potential Energy (Gravity is not Constant)

• Consider the case of throwing a ball from an initial position $r<em>1$ on Earth to far away out in space. Let the ball fall to a position $r</em>2$.

• Let’s find the work done by gravity in going from $r<em>1$ to $r</em>2$.

• We say that the work done by gravity is the area under the $F_{grav}$ vs r curve. Draw the curve.

• $W = PE<em>1 – PE</em>2 = area = \frac{GMm}{r<em>1} - \frac{GMm}{r</em>2}$

• It makes sense to say $PE<em>1 = -\frac{GMm}{r</em>1}$, and $PE<em>2 = -\frac{GMm}{r</em>2}$

• $F_g = \frac{GMm}{r^2}$

Big Idea: Universal Gravitational Potential Energy

• $PE_g$ is always less than or equal to zero.

• We do not need to set a zero line for r; it is set at infinity. Anything divided by infinity is zero.

• As the masses get farther apart, $PE_g$ increases by becoming less negative.

• Must have 2 objects to have universal potential gravitational energy.

• $PE<em>g = -\frac{Gm</em>1m_2}{r}$

• Note: This means that $PE \to 0$ as $r \to \infty$. This means that PE will be negative for all finite r’s!

• Note: You can still apply this equation to conservation of energy problem solving: Initial energy = final energy

Black Holes

Seed Question

• How fast would you have to travel in order to escape from a black hole? Is there a point of no return? Support your answer.

Spacetime

• While devising his general theory of relativity, Einstein combined the three dimensions of space and the one dimension of time into a single useful concept he called spacetime.

• All massive objects curve space and time.

• Gravity warps space and time.

• Spacetime can be thought of as an elastic sheet that bends under the weight of objects placed upon it. The more massive the object, the more spacetime bends. If the massive object is also spinning, it causes spacetime to not only bend but to twist as well.

Exploration

• The Event Horizon is the boundary between our universe and the unknown realm inside the black hole.

• The radius of the Event Horizon is called the Schwarzschild Radius $R_s$.

• Find $R_s$ for a black hole of mass M. Use the conservation of Energy. (Total Initial Energy = Total Final Energy).

• $V_f = 0$

• Travel Inside a Black Hole

Big Idea

• A black hole is a region in space where the gravitational pull is so strong that even light cannot escape. It is a great amount of matter packed into a very small area. We cannot directly observe a black hole; we can only see the radiation emitted by a black hole when it pulls in matter.

• Types of black holes:

  • Stellar: 10 to 24 times more than the mass of the sun.

  • Supermassive: More than 1 million suns together.

• The Schwarzschild Radius of a black hole is: $R_s = \frac{2GM}{c^2}$

  • Where $C$ = speed of light = $3 x 10^8 m/s$ and M = mass of black hole

Universal Gravitation

Seed Question

• What should the force of gravity between 2 objects depend on?

Exploration

• Gravitational Force between two point masses depends upon the masses of the two objects and weakens with distance.

• We can calculate the force of gravity by the Earth on the moon using $F<em>g = \frac{Gm</em>{earth}m_{moon}}{R^2}$

• Combine the Law of Universal Gravitation with Circular Motion Principles.

• The Moon orbits around the Earth. What is the $F_{net}$ for the moon? (hints: centripetal acceleration=?/?)

• $F_{net} = ma$

Solving for $v_{orbit}$

1. Solve for $v<em>{orbit}$ algebraically. Set the equation for $F</em>g$ equal to $F<em>c$ and solve for $v</em>{orbit}$.

   • Does the velocity of the moon depend on the mass of the moon? Does it depend on the mass of the Earth?

2. Use the equation for $v_{orbit}$ to calculate the orbital velocity of the Earth around the Sun. The mass of the Sun is $1.991 x 10^{30} kg$ and the distance from the Sun to the Earth is $1.496 x 10^{11} m$. $G = 6.67 x 10^{-11} \frac{Nm^2}{kg^2}$

   • a. Orbital velocity?

   • b. Calculate the period for the earth to orbit the sun. Confirm it is 365 days.

3. Communications satellites (geostationary satellites) orbit at a fixed location above the earth and must stay at a fixed altitude above the Earth. The orbital velocity of these satellites is 3070 m/s. Calculate the height of the satellite above the Earth.

   • $m<em>{earth} = 5.97 x10^{24} kg$, $R</em>{earth}=6.3714x10^6 m$

• If you don’t round the value of R above, you end up with $3.59 × 10^7 m$

Big Idea: Newton’s Law of Universal Gravitation

• $F<em>g = G \frac{m</em>1m_2}{R^2}$

• Gravitation attracts everything in the universe that has mass to everything else in the universe that has mass. This law applies between point masses, but spherical masses can be treated as though they were point masses with all their mass concentrated at their center.

• Where $G = 6.67 x 10^{-11} \frac{Nm^2}{kg^2}$ (Universal Gravitation proportionality constant). This is a measured constant and can be found experimentally.

Kepler's Laws

Kepler's Laws discuss the motion of planets. The first law states that planets travel in elliptical orbits with the sun at one of the foci. The second law explains that planets sweep out equal areas in equal time. The third law is represented by the equation $T^2/R^3 = 1$, where R is the average distance between the planet and the sun, and this constant is 1 only for objects orbiting the sun. The unit for R is in astronomical units (au), with 1 au equal to $1.4957 x 10^{11} m$.

Finding the Mass of the Earth

To determine the mass of the Earth, we can use the Moon’s orbit around it. By applying the formula $m_E = \frac{4\pi^2 r^3}{GT^2}$, and using $R = 60R_E = 60(6.4 x 10^6 m) = 3.8 x 10^8 m$ and $T = 27 days (24 hr/day)(3600 s/hr) = 2.3 x 10^6 s$, we find the mass of the Earth to be approximately $m_E = 6.0 x 10^{24} kg$.

Orbits

Kepler found that $\frac{T^2}{r^3} = constant$ for all objects orbiting the sun. Orbit problems are essentially forces problems. By sketching the orbit with $F_{grav}$ on the orbiting object m, and using the equation $F<em>{Net}$ toward center = $m</em>ac$ (where $v = \frac{2\pi r}{T}$), we derive $\frac{GMm}{r^2} = m\frac{4\pi^2 r}{T^2}$. The orbiting mass m always cancels out, allowing us to solve for the desired parameter: M, r, or T.

Universal Gravitational Potential Energy

Universal Gravitational Potential Energy involves understanding that $PE_g$ is always less than or equal to zero. With the formula $PE<em>g = -\frac{Gm</em>1m_2}{r}$, we note that $PE \to 0$ as $r \to \infty$, meaning PE will be negative for all finite r’s. As the masses get farther apart, $PE_g$ increases by becoming less negative. Two objects are required to have universal potential gravitational energy, and this equation can be applied to conservation of energy problem-solving: Initial energy = final energy.

Black Holes

A black hole is a region in space with such strong gravitational pull that even light cannot escape, characterized by a great amount of matter packed into a small area. The Schwarzschild Radius, $R_s$, defines the Event Horizon, with $R_s = \frac{2GM}{c^2}$, where $C$ is the speed of light ($3 x 10^8 m/s$) and M is the mass of the black hole.

Universal Gravitation

Newton’s Law of Universal Gravitation states that $F<em>g = G \frac{m</em>1m_2}{R^2}$. Gravitation attracts everything in the universe that has mass to everything else that has mass. Applying this, $$F_{net} = ma$