Friday, May 24, 2013

A Little News, Two Approximations, And A Few Graphical Derivatives To Go With Your Coffee

Good Morning!  While drinking my coffee after getting a full night's sleep, (hooray for happily sleeping seven month olds!!!), I came across a cute little satellite and some useful approximations.

ESTCUBE 1
The Estonian University of Tartu has successfully placed a student built and student operated cubesat into orbit.  The satellite will deploy an electrodynamic tether and test the ability of the device to propel the space craft by exploiting the force between the electric charge placed on the tether by the satellite and charged particles in the solar wind.  For those that didn't know, the electronic tether propulsion concept was patented by Robert Forward, a physicist who worked for Hughes research during the '60s and went on to become a famous scifi author[5][6].  Folks with ham radios can listen in on the satellite at 437.250 MHz and 437.505 MHz.  M5AKA did a great write up on the little cubesat[1].  The satellite tracker here at Copasetic Flow has been updated to track ESTCUBE 1.  The passes over Texas A&M for the next 24 hours are shown in picture 1.


For one of the cutest satellite deployments ever, watch the ESTCUBE 1 mission video on youtube [4].


Approximations
I'm reading up on general relativity[3] this morning for one of my research projects.  The paper zipped right through two approximations we use all the time that used to give me pause, so I thought I'd put a bit more detail here.

The first has to do with our propensity to turn quadratic equations into linear ones if the variable involved is much less than one (picture 2).



Because squaring a number that is smaller than one makes it much smaller than itself, and much small than one, after performing the binomial multiply, we just drop the squared term.

The second approximation has to do with the Taylor series for the exponential function.  Once again, assuming the argument is much smaller than one, we just jettison all the terms in the Taylor series shown below except for the first two.



Interestingly, in this case both approximations give the same result.

Graphical Derivatives
Finally, the article I'm reading mentioned that the second derivative of the absolute value of z is the Dirac delta function with z as the argument   I don't have a clue how to show this strictly mathematically, but in some class along the way, they taught us to take graphical derivatives.  The pictures 3, 4, and 5 illustrate the technique.

First, the function we're starting out with, the absolute value of z.  Note that it has a slope of -1 to the left of the origin and a slope of +1 to the right of the origin. (picture 3)


So, we can just draw out the derivative, (the slopes), based on the graph to get (picture 4)


Which we can write in terms of the Heaviside step function as shown.  Notice that it has a slope of zero on either side of the origin with an infinitely sharp slope at the origin where the value changes from -1 to +1 at 0.

The infinite slope is just represented by a Dirac delta function as shown in picture 5.


Except there's a little problem.  While you can read off the form of the derivative graphically, you have to include a little knowledge of distributional functions to get things just right.  There should be a factor of two in front of the delta function above.  The weighting factor is equal to the size of the jump in the function that the derivative is taken for.  Since ours goes from -1 to +1, the factor is 2.

For more on the relationship between Heaviside step function, the Dirac delta function, and the folks they're named after see reference 2.

References:
1.  Excellent writeup on ESTCUBE 1 from M5AKA
http://amsat-uk.org/2013/05/21/estcube-1-first-picture/

2.  Heaviside and Dirac
http://chipdesignmag.com/carter/2013/03/27/dirac-heaviside-physics-and-electrical-engineering/

3.  AJP article on general relativity
http://dx.doi.org/10.1119%2F1.2800354
Jones P., Muñoz G., Ragsdale M. & Singleton D. (2008). The general relativistic infinite plane, American Journal of Physics, 76 (1) 73. DOI:

Open access version:
http://arxiv.org/abs/0708.2906v1

4.  ESTCUBE 1 mission video
http://youtu.be/2EUIsUPzwaE

5.  Forwards patent
http://www.google.com/patents/US6116544

6.  More on Forward
http://copaseticflow.blogspot.com/2013/05/gravitomagnetism-and-antigravity-for.html

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