Day 4 Orbits and Gravity OpenStax Astronomy Ch. 3

Johannes Kepler and the Laws of Planetary Motion

Kepler used decades of Tycho’s observations in his mathematical calculations, to determine the shape of the planetary orbits, and the speed of the planets as they went around the Sun. This massive effort resulted in three major statements about the characteristics of planetary orbits: Kepler’s three laws of planetary motion.

Kepler’s three laws of planetary motion Orbital paths of the planets are ellipses. An imaginary line connecting the planet with the Sun sweeps out equal areas of the ellipse in equal intervals of time. The square of a planet’s orbital period is proportional to the cube of its semi-major axis. Kepler published this in 1609, the same year that Galileo built his first telescope.

Kepler’s laws of planetary motion Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time. Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

An Ellipse can be drawn with string and TWO foci

For an ellipse, r1 + r2 = 2a The eccentricity is defined as: e = c/a A circle results when e = 0 GeoGebra demonstration: http://people.ucalgary.ca/~louro/geogebra/ellipse.html

Some Properties of Planetary Orbits

Key terms for orbits Perihelion – the point in the orbit that is closest (peri-) to the Sun (helios). Aphelion – the point in the orbit that is furthest from the Sun. Then for the orbits of the Moon and spacecraft: Perigee – the point in an orbit around the Earth that is closest to Earth. Apogee – the point in an orbit around the Earth that is furthest from Earth. Semimajor axis – half (semi) of the longest (major) line (axis) across the ellipse

Kepler’s laws of planetary motion Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time. Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

Kepler’s Second Law: equal areas in equal time This also means higher speed at closer distances.

Another graphic on Kepler’s Second Law:

The Astronomical Unit is about 150,000,000 km

Kepler’s laws of planetary motion Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time. Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

Kepler’s Third Law: P2 (in years) = a3 (in a. u Kepler’s Third Law: P2 (in years) = a3 (in a.u.) Basically, it means that large orbits have long periods. The data are easier to understand if we use years and A.U.

Let’s review Kepler’s Laws Let’s review Kepler’s Laws. Review:  see if you can tell what these are simulating:  http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_kepler2.html http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_periods_sim.html http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_solar_sim.html

Newton’s Laws of Physics First law: inertia Second law: F = ma or acceleration = force / mass Third law: Action and Reaction means that forces occur in pairs. These can be used to show that orbits should obey Kepler’s 3 laws.

Isaac Newton developed a quantitative and explanatory theory of mechanics, explaining the motion of objects resulting from forces.

Newton’s First Law: The law of inertia. conservation of momentum An object will continue in it’s motion without change of velocity unless it is acted on by a net external force. This was not obvious to people before we had low-friction bearings, wheels, etc. You have seen objects in a spaceship cabin “float” through the air. This law is also related to the principle of conservation of momentum (the product of mass and velocity, p=mv)

A stimulus-response form of the 2nd law is a = F/m. Newton’s Second Law: F = ma The acceleration of a mass is proportional to the total force acting upon it, and inversely proportional to the mass of the object. Acceleration is a change in velocity (either speed or direction) measured in meters per second per second. A larger force will cause a given mass to accelerate more. Conversely, a larger mass will accelerate less under the influence of a given force. A stimulus-response form of the 2nd law is a = F/m. The “cause” is F on the right side of the equation, and the “effect” is the acceleration on the left side.

Newton’s Third Law: Action-reaction For every force acting upon an object (action), there is a force acting on another object (reaction) which has the same magnitude (size) but points (acts) in the opposite direction. Again, this is not so obvious. An example is contact forces. Another is the reaction to a rocket exhaust. In the case of gravity, both objects experience a force.

Real orbits have the center of mass as one focus For the Sun and planets, this is not a large effect. For binary stars, the center of mass may be near the middle of the line connecting them.

Newton also developed the universal law of gravity Newton also developed the universal law of gravity. Gravitational force varies with the distance between the objects. It depends on the product of the two masses, i.e., m1 x m2 and on the inverse of the square of the distance between the masses (assuming they are small compared with the distance). 1/r2

The Sun’s gravity causes planets to move on a path called an orbit The Sun’s gravity causes planets to move on a path called an orbit. These orbits obey Kepler’s Laws.

Center of mass For two objects with masses M1 and M2, the center of mass is between them, but closer to the larger mass. This means that the Sun wobbles as the planets revolve around it. This is also true of any exoplanets orbiting around a distant star (we will come back to this idea later). The star’s motion can indicate that it has planets around it.

Newton’s Laws explain Kepler’s Laws Newton’s Laws account for all three of Kepler’s Laws. The orbits of the planets are ellipses, but it is also possible to have orbits which are parabolas or hyperbolas. (conic sections) The shape of these orbits is a consequence of the 1/r2 form of gravity. Edmond Halley predicted a comet would return in 1758 and every 76 years after that. (seen in 1910, 1986, and will return in 2061) Halley’s comet has an elliptical orbit extending out past Neptune. William Herschel discovered Uranus in 1781 by accident. After 50 years it was seen to deviate from an elliptical orbit, and a calculation led to the discovery of Neptune in 1846. To be precise, elliptical orbits would only occur if there were only the Sun and one planet. There are 8 planets and other objects which cause deviations from the perfect elliptical orbit.

Kepler’s 2nd law is due to conservation of angular momentum For a revolving object, the angular momentum is L = mvr where m is the mass of the object, v is its speed (velocity v, in a circular orbit), and r is the distance from the axis of rotation or revolution. Torque The “twisting force” that changes angular momentum is called torque. Without any torque, the angular momentum of a rotating object stays the same. We can demonstrate this with bike wheels and a rotating stool. With no torque, we get conservation of angular momentum.

Kepler’s Second Law: equal areas in equal time This also means higher speed (v) at closer distances (r).

The first exam is on Thursday, Sept. 7 (this week The first exam is on Thursday, Sept. 7 (this week!) We will have about 30 minutes of class before the exam. Then you will take the exam (which uses a Scantron). The exam is multiple choice and true/false questions. Coverage is Chapters 1 to 4 in your textbook. To review, look at the chapter summaries, “key terms”, my day notes, and a study guide that I will post this weekend.