Interstellar Space

I just found this really interesting article about the Voyager Interstellar Mission which relies on two spacecrafts sent from Earth back in 1977. Check it out: http://voyager.jpl.nasa.gov/mission/

The Paradoxicality

Newton v. Einstein

As mentioned in Death by Black Hole by Neil deGrasse Tyson, both Newton's Second Law of Motion (F=ma) and Einstein's Special Theory of Relativity accurately explain motion in certain environments. Newton's law states that force is equal to the product of the mass and acceleration of an object. This explains most motion here on Earth however Newton assumed that mass was constant. But what if it's not? What happens to a spaceship that constantly loses mass as it burns fuel? What if the mass of an object somehow changes without adding or subtracting matter? This is where Einstein takes over. His famous Special Theory of Relativity describes an object as having an unchanging "rest mass" to which one adds more mass according to the object's speed. When an object accelerates in Einstein's relativistic universe, its resistance to that acceleration increases (showing up in the equation as an increase in the object's mass). So Newton said mass is unchanging and Einstein said it could change. Who was right? As far as I know, Newton's law is ideal in environments with relatively weak gravity like we experience on Earth and Einstein's Special Theory is ideal in extremely high gravity environments such as Mercury, being the closest planet to the Sun and thus experiencing the strongest gravitational pull toward it. Since Einstein's "relativistic" effects only become significant when the object approaches the speed of  light, Newton could not have possibly known about them because the study of the speed of light was after his time.

Lagrange Points of the Earth-Moon System

In Death by Black Hole by Neil Degrasse Tyson, I just read about spots called Lagrange points in which the gravitational and centrifugal forces between two bodies balance and thus basically cancel each other out. As calculated by Joseph-Louis Lagrange, the first spot (termed L1) of the Earth-Moon system falls between the Earth and the Moon, slightly closer to Earth than the point at which the Earth and Moon's gravitational forces (excluding centrifugal) balance precisely. In other words, closer to the Moon than the Earth. At this point, any shift sideways will cause the object to lose the balance and thus plummet to either the Earth or the Moon. L2 lies on the Earth-Moon system and exists on the far side of the moon while L3 exists on the side of the Earth opposite the Moon. At these two points, objects can orbit Earth with the same monthly orbit of the Moon. The balances of these two point are less precarious because it takes only a little fuel to reenter the orbit. L4 and L5, however are much more useful. Lying on either the far left or the far right side of the Moon, each point forms an equilateral triangle with the centers of the Earth and Moon. That being said, despite drifting in any direction an object can remain in orbit. Raw materials there would be under very little stress from gravity and the orbits are very stable making L4 and L5 ideal spots for space control centers.These points are occupied by satellites such as the Hubble Space Telescope and the International Space Station. Lagrangian points exist for all two-body planets. Tyson suggests that since "interplanetary trajectories that begin at Lagrangian points require very little fuel to reach other Lagrangian points or even other planets", these points can eventually be used as fueling stations to allow for more extensive space exploration.

The Paradoxicality

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Fractals and Coastlines

So I just started a book called Death by Black Hole by Neil DeGrasse Tyson and he mentioned fractals, which I had heard of before but I never looked into it. Tyson said that in 1967 Benoit B. Mandelbrot posed a question asking for the length of the British coastline. That seems simple enough. Sure, Mandelbrot couldn’t just google it but he could find a map of the coastline, measure the coastline with a piece of string, and find the length using the map’s scale. However, an issue arose. As he measured the coast on maps of increasing detail he saw that the length was increasing. That’s odd. So what was the true length? Even if one was to walk along the coast with miles worth of string going around each and every rock on the beach, it would still not be the true length because one can go to the atomic level and measure around each atom. This mental exercise evolved into a new field of mathematics based on fractional (or fractal) dimensions. Mandelbrot said that the ordinary concepts of dimension are too simple to characterize the complexity of the coastline. Fractals are ideal when describing ‘self-similar’ patterns, being mainly the same at different scales. Although the perfect fractal can only be created by a computer, they can be found in broccoli, ferns, and snowflakes.

The Paradoxicality

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The Speed of Light

So today I thought I would talk a little about the history speed of light which I read about in Simon Singh's Big Bang. In the late 1600s one of the biggest questions for scientists was whether light had a finite or infinite speed. Although the solution is usually attributed to Ole Rømer, Jean-Dominique Cassini first proposed the idea that the speed of light was finite. You may be wondering how we could have possibly proven whether such a statement is correct. Well Cassini and Rømer did what many scientists of their age did--they looked to the sky. They studied the movements of Jupiter's moon Io which appeared (from Earth) to have an irregular orbit; Io's half moon was often several minutes before or behind its predicted appearance. What Rømer realized was that this irregularity could be explained if only the speed of light were finite. Since the Earth orbits the Sun 12x faster that Jupiter, Jupiter hardly moves while the Earth moves significantly. This means that sometimes Jupiter and Earth are at opposite sides of Earth's orbit and thus much further apart. If the speed of light had a finite value, then the significant variations in distance between Io and Earth would account for the half moon being early or late because the light would take more time to travel the extra distance when the two planets are at their furthest points. So Rømer spent three years recording the observed timings of Io and the relative positions of Earth and Jupiter. He then estimated that the speed of light had a finite value of 190,000 km/s (it is actually 300,000 km/s). Although somewhat inaccurate, this value was close enough that he accurately predicted that an eclipse of Io on November 9 1670 would be ten minutes late, thus proving that the speed of light was indeed finite.

The Paradoxicality

The Sun is Yellow Because the Sky is Blue

I was reading in Strange Universe by Bob Berman about the colors we perceive and learned something I hadn't known before. All the colors that we see on Earth do not actually originate here at all. They are a result of the colored photons created in the center of the sun. Our sun is actually colored pure white but radiates the colors of the spectrum. Around sunset the sunlight refracts, or bends, due to the varying thicknesses of the atmosphere and separates into individual colors of the spectrum side by side creating the beautiful background that we get to witness.
    The sky is blue because the shorter blue wavelengths of the sun's light are more easily scattered by the air molecules in Earth's atmosphere than the wavelengths of other colors. This causes the atmosphere, heavily populated by air molecules, to appear blue. As a result of the blue sky, the sunlight we see is actually missing some of its blue wavelengths. The sun thus appears yellow instead of white. However there is a way to see the true color of the sun that is seen by astronauts when above Earth's atmosphere. When light illuminates a white surface, such as snow, the two mixtures of light combine (the blue wavelengths from the  blue sky and the rest of the spectrum from the yellow sun). The snow thus becomes a mixture of the two colors and appears to our eyes as white.

The Paradoxicality

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The Monty Hall Problem

One problem that I have read about in several books involving probability is somewhat popular and is known as the Monty Hall problem. It caused much debate in the 1990s and is described by the following:
    
"The set-up is a standard game show in which the contestant is trying to win a prize secreted behind one of three closed doors. Behind two of the doors is an amusing but essentially worthless prize, like a sheep, while the third conceals something desirable, like a Mercedes. When the contestant chooses a door, the game-show host quickly opens one of the two remaining doors and reveals a booby prize. The host then offers the contestant a choice: Either stay with the original pick or switch to the other closed door. That's the situation. The question: Is it in the contestant's best interest to switch or does it not matter?" ( Strange Universe by Bob Berman). Note: Monty Hall, the host, knows which door the desirable prize is behind.

The solution is somewhat counter-intuitive and was highly debated among mathematicians. My first thought was that it didn't matter and that each door has the same 1/3 chance in containing the Mercedes. However it is in the contestant's best interest to switch to the other door. Why? Switching gives the contestant a 2/3 chance of winning whereas keeping the original door only gives a 1/3 chance. Let me explain. Originally there is a 1/3 chance that the contestant's first choice, lets say Door A, is the winning door. There is a 2/3 chance that either Door B or Door C is the correct door.
So Door A is set 1 and Door B and C are set 2. The odds say that the prize is more likely to be in set 2 however the contestant cannot pick 'set 2' they must choose a specific door. Then Monty Hall opens Door B in set 2 which he knows does not contain the prize. This leaves Door C as the only door left in set 2. Since there is still a 1/3 chance of the winning door being in set 1 and a 2/3 chance in the door being in set 2, the contestant is more likely to win by switching to the other unopened door, in this case Door C. This solution has been tested and supported with computer simulations of a similar scenario.

Let me know in the comments if I explained it correctly. Have you heard of the Monty Hall problem? Do you agree with the solution? Does it even make sense?

The Paradoxicality