Today’s topic: Some Celestial Mechanics F Numeriska beräkningar i Naturvetenskap och Teknik

Coordinate systems Cartesian coordinates Unit vectors are orthogonal with norm 1 Numeriska beräkningar i Naturvetenskap och Teknik

Cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik

Vector- och scalar product in cylindrical coordinates Orthogonal Right hand system Numeriska beräkningar i Naturvetenskap och Teknik

Spherical coordinates Numeriska beräkningar i Naturvetenskap och Teknik

Force law Torque Angular momentum gives: Introductory mechanics Numeriska beräkningar i Naturvetenskap och Teknik

The angular momentum is constant in a central force field... Numeriska beräkningar i Naturvetenskap och Teknik A quantity that does not change with time, i.e. our case does not change along the trajectory of a planet is called a: CONSTANT OF MOTION If we can find a quantity whose time derivative is zero that quantity is a constant of motion.

Central force r x p is orthogonal to r, i.e. r is orthogonal to L which is constant. 1 Angular momentum is a constant of motion 2. Motion is in a plane Numeriska beräkningar i Naturvetenskap och Teknik

In order write down the equations of motion we need the acceleration in cylindrical coordinates. This problem relies on the calculus you learn in math class! Numeriska beräkningar i Naturvetenskap och Teknik

Velocity in cylindrical coordinates Motion in the plane due to central force Radial velocity Angular velocity Numeriska beräkningar i Naturvetenskap och Teknik

In the same way… acceleration in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik and also using the same method we can derive

Numeriska beräkningar i Naturvetenskap och Teknik Acceleration in cylindrical coordinates Look at this at home!

Acceleration in cylindrical coordinates Ins. from above gives that we have TWO components Numeriska beräkningar i Naturvetenskap och Teknik

Equations of motion in the central force system this can also be written as: with the acceleration in the plane Numeriska beräkningar i Naturvetenskap och Teknik

Equations of motion in the plane in cylindrical coordinates Depends explicitly on the force Can be integrated without defining F Now, we note that i.e. Which gives Numeriska beräkningar i Naturvetenskap och Teknik

Sector velocity Kepler’s second law Numeriska beräkningar i Naturvetenskap och Teknik

Rho direction: Equations of motion in the plane in cylindrical coordinates The angular momentum can be used to switch between rho and phi! sinceWe have Substitution gives: We have two functions of time, rho and phi. We want ONE! Numeriska beräkningar i Naturvetenskap och Teknik

The energy is a second constant of motion... Numeriska beräkningar i Naturvetenskap och Teknik

A second constant of motion A conservative force, i.e. a force with potential Numeriska beräkningar i Naturvetenskap och Teknik WHY of interest? Examples of such forces?

A second constant of motion Numeriska beräkningar i Naturvetenskap och Teknik How to get a first order time derivative out of this?

A second constant of motion We think of the chain rule again and multiply by These are equal Numeriska beräkningar i Naturvetenskap och Teknik

in the eq. below We now have time derivatives on both sides of this equation! i.e. Continue by looking at the left hand side l.h can be written Numeriska beräkningar i Naturvetenskap och Teknik Look at this at home!

Kinetic energy from radial motion From L constant we have (still) Numeriska beräkningar i Naturvetenskap och Teknik Potential energy Kinetic energy from motion in phi Lets identify the terms!

Solving the equations of motion One can now either try to integrate with respect to the time, t, or, one can solve with respect to the angle. Two steps for a ”straightforward” solution. 1.Transform equation to be distance rho as function of the angle phi instead of time. 2. Make a 1/rho substitution to create a standard linear diff. eq. with constant coefficients. Numeriska beräkningar i Naturvetenskap och Teknik

Solving the equations of motion From second order time derivative to second order derivative in phi: Numeriska beräkningar i Naturvetenskap och Teknik Apply it two times

At this point we have but Binet! Numeriska beräkningar i Naturvetenskap och Teknik

Solving the equations of motion Binet’s equation for the kepler case (1/r 2 ) Second order diff equation. (solve with characteristic equation!) Numeriska beräkningar i Naturvetenskap och Teknik

Different orbits Reference direction when α is zero Numeriska beräkningar i Naturvetenskap och Teknik

Different orbits Investigate in the project! Numeriska beräkningar i Naturvetenskap och Teknik

Other thoughts: Which velocity, in which direction, will give circular orbit? Is there a maximum velocity for a planet to stay in a closed orbit around the Sun? If the velocity is below the escape velocity, how does different start angles influence the shape of the orbit? Can you create ellipses and circles from the same starting speed? If a small planet passes close by another planet (e.g. an elliptic orbit that passes close to a jupiter like planet) what will happen. Why? (Voyager slingshots). If we integrate what should be a closed orbit with bad precision what will happen? Numeriska beräkningar i Naturvetenskap och Teknik

Orbital motionρ(t) Numeriska beräkningar i Naturvetenskap och Teknik

Orbital motionρ(t) This integral can in principle be solved t(ρ) but its inversion ρ(t) is not possible in ”simple functions”. The same is true for the angle as a function of time. Numeriska beräkningar i Naturvetenskap och Teknik

Extra Numeriska beräkningar i Naturvetenskap och Teknik

Mean anomaly (ohmega constant, if e=0) Actual angle = true anomaly) Variable substitution... Half major axis Eccentric anomaly Numeriska beräkningar i Naturvetenskap och Teknik

After this substitution... Kepler’s third law (can also be found from geometrical considerations) Numeriska beräkningar i Naturvetenskap och Teknik

Kepler’s equation How find ρ(t)? Only numerical solution Gives ρ for this t! Generally at time t Numeriska beräkningar i Naturvetenskap och Teknik

Two body problem For two interacting bodies the mass above is substituted by the so-called reduced mass Three body problem... Many tried to solve it (Poincare and others) but no solution exists in simple analytical form. Power series expansions exist. The problem has a very interesting background story. As an example, find and read on your own the story behind the Mittag-Leffler prize. Numeriska beräkningar i Naturvetenskap och Teknik

Notera att volymelementet i cylinderkoordinater är: Numeriska beräkningar i Naturvetenskap och Teknik