Application of the Radiating Orbital Equation

Please read my previous blog post, Deriving Kepler's Laws for Radiating Stars. In this blog post, we take the previously mentioned ideas and apply them to see if it would ever be necessary in calculation. Read about it here.

Deriving Kepler's Laws for Radiating Stars

In this next blog, I examine how Kepler's Laws (well, just the first one at least) changes when we look at a variable-mass system. Read about it here.

Eccentricity given mass, distance, and relative velocity

Suppose we have two stars; we know their masses, distance from each other, and relative velocity between them. We seek to predict the eccentricity of their orbit.

Read about it here.

Making Mistakes in Critical Thinking

I spent a great deal of time focusing on a simple problem in my recent midterm. I knew the correct answer, but sought a "more correct" answer. Read about it here.

Space Elevators: Exactly what it sounds like

There was a bit of math I had to write down, so please enjoy the PDF version here.

Star Wars - The Kessel Run in under twelve parsecs

"You've never heard of the Millennium Falcon? It's the ship that made the Kessel Run in less than twelve parsecs." - Han Solo, Star Wars: Episode IV A New Hope

That certainly sounds impressive, I think... What exactly does that mean though? Anyone reading this blog probably already knows that a parsec (pc) is a measurement of distance, not time or speed. A parsec (parallax arcsecond) is the distance an object would need to be to create a parallax angle of one arcsecond when the base between the two points of observation is an astronomical unit (average Earth to Sun distance) apart.

Most people with this much information will simply assume the George Lucas was just making up science fiction sentences and put no extra thought into that line, and I would be one to agree. But that is not why we are here. We are here because there is still an air of curiosity in that claim that is worth investigating, for no other reason that it is somewhat interesting.

Supporters of George Lucas will claim that this statement still has some validity when you consider relativistic effects. At speeds near the speed of light, the phenomenon of length contraction occurs, as does time dilation.

One of the key points of relativity is that, no matter which reference frame one is in, the speed of light is always the same and no physical object can exceed this speed. One of the consequences of this fact is that, if you are moving in respect to some object, that object will appear "thinner" or contracted if you looked at it. The relationship between these two properties is

where L_0 is the length of the object in its reference frame, L is the length of the object in your reference, v is the speed you are going, and c is speed of light. If we look, we see that v cannot exceed v, otherwise we would get an unreal number.

So why is this important? Couldn't Han Solo have just said the Millennium Falcon went at some speed v? Or made the Kessel Run in some time t? To answer these questions, let us look at an important fact, the change in speeds vs. the change in lengths. Assume the Millennium Falcon went at some speed v=0.9999c while making the Kessel Run in 12 pc. Solving for L_0, the proper length of the Kessel Run, we get 849 pc.

The significance of this is how much the speed changes versus how much the distance changes. If a regular ship could only make the run at v=0.9995c, then the relative length of the Kessel Run would be 26.83 pc. The difference in speeds is only 0.04%, whereas the difference in lengths is 124%! 

When it comes to relaying information to another human being, bigger changes grab more attention. Could this be the reason Han Solo toted the Falcon's speed with a measure of distance rather than time? I doubt George Lucas and his writers had originally thought so deeply about this line (despite them saying they did after the fact), but it's definitely an interesting way to say how fast something is when you move into the realm of relativity.

What causes the seasons?

In this brief paper, we look at the what causes the seasons. Specifically, why is it hot in the summer and cold in the winter?

The primary source of heat for the surface of earth is the Sun. The Sun is a giant thermonuclear fusion reactor, releasing large amounts of energy as a byproduct of its fusion reactions. Let us assume that the Sun is perfectly spherical. Let us also assume that its surface temperature is uniform (ignoring sun spots and solar flares). Then the energy radiation will be isotropic and traveling away from the sun. Because this emitted energy is conserved, all the energy on the surface of some sphere of radius r_1 will be the same on some other sphere of radius r_2 . Therefore, we conclude that the energy per unit area is proportional to the square of the distance from the Sun.

To relate this to temperature, we posit that the temperature of an object increases with the more energy per unit area it receives in a given area. For a flat surface, this is proportional to the flux of the energy through the surface.

 

T∝E∙s

Where T  is the temperature at some point, E is a vector describing the direction of energy travel and the amount of energy and s  is the surface normal to the point of energy capture.

  

The mean Earth-Sun distance is D_S=1.5E8 km whereas the radius of the earth is only R_E=6.37E3 km. We see that D_S/R_E>>1 and can approximate the energy propagation as constant in direction in the region of the earth. Further, earth has aphelion D_A=1.52E8 km  and perihelion D_P=1.52E8 km. This time, we see that D_A/D_P=1.03 . There’s about a 3% difference, but is that enough to account for the drastic temperature change throughout the seasons? 

Another important factor to consider is the tilt of the Earth. The rotation of the Earth actually happens at tile of about 23.4° compared to the plane of its orbit. This tilt always points in the same direction relative to the stars. Therefore, a location on Earth can be directly above the Sun at one moment in time, but only get 66.6°  of its impact in another season. This change is about cos 0°/cos 23.4°. This is a least-case scenario. If we went 60° N , then we can get a change of about cos 60°/cos 83.4°=4.35. This is a much more dramatic change compared to Earth’s proximity to the Sun.