Wednesday August 31 2011

Rotation of Moon.. The fact that we always see the same side of Moon means that it must rotate on its axis once for every orbit around the Earth. This is called synchronous rotation and is due to tidal locking. So the Moon, as seen from Earth, appears NOT to rotate, but as viewed from an outside vantage point it rotates about once a month. The true rotation period is that seen by an outside observer, and is often called the sidereal period (sidereal= relative to stars).

Potential Energy (PE). Astronomers put the zeropoint of potential energy of two bodies at "infinity". Consider two masses initially at "infinity" (very far apart) and initially at rest. At this point, the kinetic energy (KE) is 0 (at rest) and the PE is 0 (by definition of PE zeropoint). So the total orbital energy is 0. As time goes on, the particles will start moving towards each other by their mutual gravitaional attraction. The force will increase as the particles get closer, and their speeds will increase. Thus, the KE (green curve) must increase with time. By conservation of energy, the TOE must remain 0 at all times, so as the KE goes up, the PE *MUST* go DOWN (pink curve). By integrating the force of gravity with distance, we can find the work done by gravity, which gives us the KE as function of the distance between the bodies. The PE must be the negative of that.

Just Say NO to mgh!. For orbital problems, the PE is not equal to mgh! mgh may be a useful equation if you are figuring the speed of a roller coaster or building a bridge, but is not useful for doing orbital problems in astronomy. These slogans are takeoffs of the anti-drug slogans that we students heard in the 1960s.

Escape velocity. If we shoot a rocket from the surface of the Earth with the escape velocity, the rocket will travel out into space and not return. The rocket will continuously slow down, but (in an idealized case) reach zero speed at an infinite distance.

KE and PE in circular orbit. As an object orbits the Sun in a circular orbit, its speed and hence its KE (kinetic energy) are constant. Equations are given for the KE and PE of the low mass orbiting body. (The KE equation can be easily derived from Kepler III). The total energy is a constant that is always negative, as this is a bound orbit.

Some parameters of an ellipse. This shows the semimajor axis (a), the semiminor axis (b), and the definition of the eccentricity (e). We can specify the "squashedness" of the ellipse with the ratio of the semiminor to semimajor axis (b/a) or the eccentricty e. A little trig shows the relationship between b/a and e: b/a=sqrt(1 - e**2). This particular ellipse has e=0.6 and b/a=0.8. A circle has b/a = 1 and e=0.

KE and PE in elliptical orbit. As an object orbits the Sun in an elliptical orbit, its speed and hence its KE (kinetic energy) constantly changes. The object speeds up as it "falls towards the Sun" moving from aphelion to perihelion, then slows as it "coasts uphill away from Sun" from perihelion to aphelion. Note that the object does NOT stop moving at aphelion- the KE is always a positive, non-zero value. (If the object ever stopped moving with respect to Sun, it would fall straight towards center of Sun and collide with it.) Note that the PE and KE curves are mirror images of each other. KE + PE sum to the total orbital energy, which has a constant value less than zero (because this is a bound orbit).

Speed in elliptical orbits. Here are 2 orbits that have the same value of semimajor axis (a). One is circular, one elliptical (with b/a = 0.5 or e = 0.866). As both orbits have the same a, they would have the same orbital period (P) around the Sun. The object on the circular orbit would have a constant speed (by symmetry- there is nothing to tell one part of orbit from another). The object in the elliptical orbit would have a continuously varying speed, fastest at perihelion (point on orbit closest to Sun), slowest at aphelion (farthest from Sun).

The TOTAL ORBITAL ENERGY (TE) depends ONLY on a, so equal mass objects would have the SAME *TE* in the circular and the elliptical orbits. Thus, the object in the ellipitcal orbit would have speed equal to the circular orbit speed as it crossed the circular orbit distance from the Sun, as pointed out on diagram.

Different shaped elliptical orbits. If we start out our rocket/cannonball from the same starting point, with the same KE (same speed),but with various angles to the tanget to the surface of the Earth, we get orbits with the same values of a (semimajor axis) but with different values of e (eccentrcity). All these orbits have the same TOE.

Launching the rocket "straight up" would result in it moving in a straight line (ignoring rotation of Earth). This is just a degenerate ellipse orbit.

Hohmann or least energy orbits. With the usual "kick and coast" rockets, the Hohmann or least energy orbits are the paths that you would take to get to another planet using the least amount of energy. (Hohmann was a German scientist who published this idea in 1925.) The basic idea is that you want to get just to the distance of the second body from the Sun, and no farther (or closer for interior planet), as you need energy to move away from the Sun (or towards Sun for interior planet!)

Since the Hohmann orbit is just an ellipse with known aphelion and perihelion, you can easily find the semimajor axis and from that the period of the Hohmann orbit- it obviously takes 1/2 the period to get from the inital body to the other body.

Now, once you get to the other body, you will not be going at its speed, so you will have to use energy to change your speed to match the destination body.

The speeds are given in miles/second, not meters/second (sorry!). If you are going from Earth to Mars, you would start out with the Earth's orbital speed (18.5 miles/sec) and have to add speed to get to the Hohmann orbit (you need 20.4 miles/second at the perihelion of the Hohmann orbit). Of course, if you were really leaving from the surface of the Earth, or even Earth-orbit, you would need additional energy to get away from the Earth's gravitational pull.

Conic section curves. The intersection of a plane with a cone will generate curves called conic sections. Gravitational orbits result in paths which are always a conic section curve- ellipses (circle is a special ellipse) for bound orbits, parabola for the dividing line between bound and unbound orbits, and hyperbolas for unbound orbits.

KE and PE in parabolic orbit. This is an orbit with a total orbital energy of zero. In such an orbit (in the idealized case) the object will not reach a finite distance and turn around. Thus we say this is an unbound orbit. All circular and elliptical orbits have a "turn around point" so they are bound orbits. It is useful to consider the parabola as the dividing line between bound (elliptical) and unbound (hyperbola) orbits. Bound and unbound are also related to TOE. All bound orbits have a negative TOE - all unbound orbits have a positive TOE. Parabolic orbits, with TOE = 0 , mark the dividing line between bound and unbound.

KE and PE in hyperbolic orbit. This is an unbound orbit with a positive total orbital energy.

Conic section curves and orbits. The intersection of a plane with a cone will generate curves called conic sections. Gravitational orbits result in paths which are always a conic section curve- ellipses (circle is a special ellipse) for bound orbits, parabola for the dividing line between bound and unbound orbits, and hyperbolas for unbound orbits.

The shape of the orbit (ellipse, parabola or hyperbola) is also related to the total orbital energy (TOE). All ellipses have TOE less than zero. All parabolic orbits have TOE equal to zero. All hyperbolic orbits have TOE greater than zero.

Newton thought experiment. This is basically same as the previous Newton thought diagram, but with a little more detail. The curves labeled A to G show the effects of increasing speed for the projectile. A shows a speed significantly less than the circular speed- this results in an elliptical orbit that intersects the Earth. (See also Fig 4.5). The portions of the orbit which are external to the Earth are indeed the same as an elliptical orbit around a point mass at the Earth's center. B shows an (elliptical) orbit resulting from an inital speed which is only slightly less than the circular orbit at the initial height. C shows the circular orbital speed. D shows an initial speed greater than the circular speed, but less than the escape speed, resulting in an elliptical orbit. E shows part of the path of an object launched with just the escape speed- it is a parabola. F shows the path of an object launched with somewhat more than the escape speed. G shows the path of an object launched with many times the escape speed. F and G paths are both hyperbolas and represent unbound orbits- the projectile will never return to the vicinty of the Earth.


The following show just a few examples of the rich phenomena that are possible when we have more than 2 bodies involved in gravitational interactions.

General three body problem. If one starts with TWO bodies of arbitrary masses, initial distance, and velocities, you can easily figure out the future orbit of the bodies. It will be an ellipse, parabola, or hyperbola, depending on the initial TOE. (In the general case the equations are a little more complicated than we have considered, as we have to worry about motions of both bodies around the center of mass of system, but this is not much of a complication.)

However, if we add ONE MORE BODY we cannot write down equations that would specify the future positions of the bodies! The best mathematicians in the world have looked at this problem for centuries and no one has found a general solution. (There are approximate solutions for special cases , say where there is one very massive body and two very light bodies.)

So how can we deal with the real Solar System (or real galaxy, or real universe) which obviously has more than 2 bodies? We can use a computer to make a numerical simulation of the motion of the bodies. Such simulations are usually called "n body simulations", where n is the number of bodies in the simulation. We start out with initial positions and velocities. We assume the velocities (speeds and directions) are fixed for some small time step, and we calculate the new positions at the end of the small time step, using the initial positions and velocities. Now we use the new positions to calculate the (changed) gravitational forces between the bodies, from which we can calculate new velocities, so we start over again. Essentially, we need to assume the objects move in a straight line over the time step. (The real paths will be curved). If we have a small enough time step, we can follow the motion to high accuracy. Small time steps= MANY calculations, which is why dynamicists (people who study the dynamics or motions of gravitating systems) are always wanting faster and faster computers.

Of course, computational dynamicists have invented many numerical and programing "tricks" to speed up such computations, but they still will gladly use all the CPU cycles they can beg, borrow, or steal.

Galileo spacecraft path. The Galileo spacecraft was sent to Jupiter by first stealing some energy from the Venus-Sun system, then stealing some energy from the Earth-Sun system (twice!).

Voyager2 speed. The Voyager 2 spacecraft stole energy from Jupiter-Sun, Saturn-Sun, and Uranus-Sun systems! Note the "spike" in Voyager 2 speed at each planet. The craft sped up at it fell towards the planet, then slowed down as it passed by the planet (and had to "coast uphill" away from the planet), but, due to the stolen energy, the speed at which the craft left the vicinity of each planet was greater than the speed as it approached the planet. Voyager 2 is now leaving the solar system, having acheived a speed greater than the escape speed from the Sun.

Where does the "stolen" energy come from that speeds up the spacecraft? NOT from the KE of Jupiter!! In fact Jupiter ALSO SPEEDS UP in its orbit! Jupiter gets closer to Sun, so the PE of the Sun-Jupiter system gets smaller (more negative). So the energy to speed up spacecraft AND Jupiter comes from PE of Sun-Jupiter system. (Of course, the fractional change of speed of Jupiter is incredibly small, as Jupiter has a mass about 24 powers of 10 times the mass of Voyager 2!)

Comet path altered by Jupiter. In this cartoon, a comet on a parabolic orbit (with zero TOE) interacts with Jupiter to end up in an elliptical orbit (with negative TOE). Thus, the Jupiter-Sun system would GAIN the energy lost by the comet. Jupiter would move minutely away from Sun and slow down minutely in its orbit. Whether such an interaction cause the small body to gain or lose energy depends on the details of the interaction- the angle between the velocities of Jupiter and the small body, whether the body comes "in front of" or "behind" Jupiter as it orbits the Sun, etc.

MESSENGER trajectory. Spacecraft MESSENGER, now on its way to Mercury, is taking a very long and complicated path to Mercury, using multiple gravity assists.

Gravitational slingshot. An advanced race could accelerate spacecraft to enormous speeds by stealing energy from a binary star system- this is same physics used to speed up Voyager (and other interplanetary probes launched by the primitive Terran civilization in galaxy quadrant Z123-72), but on a much grander scale.

The best systems for such a "gravitational slingshot" would be systems where at least one component was a stellar-mass "compact object" - white dwarf, neutron star or black hole- so that the craft could get as close as possible to the center of one of the objects without crashing into it. Black hole binaries would be the ultimate objects to steal gravitational energy from!

What would happen to the binary star system after the craft "stole" energy? The stars would SPEED UP in their orbit, and would get CLOSER TOGETHER. This at first seems backwards- naively you would expect the stars to slow down as the system loses energy- but it makes perfect sense when you factor in the potential energy (PE) - the stars lose more PE when they come closer together than they gain kinetic energy (KE) when they speed up -so the total energy of the star system after the encounter is LOWER than before- as it must be, as the craft removes some energy from the star system, just like the Sun-Jupiter system had a little less energy after Voyager 2 used some energy to speed itself up.