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### Help with the math behind the 14 point star

Posted: Tue Jul 21, 2015 6:42 pm
Hello Guys,

In the Navy Paper they mention that the cusp confined plasma volume is 5000 cm^3. How far away then, would each point of the confined volume be from the center point? We need the mathematics to describe this object. It is a 14 point star. I attempted to draw it out:

I am pretty sure this is a "constant" style equation, where we square or cube X, Y and Z values and set them equal to a constant. That should produce a surface like the one pictured.

### Re: Help with the math behind the 14 point star

Posted: Tue Jul 21, 2015 11:24 pm
Perhaps, and I repeat perhaps, you are overthinking it. would have an average with At low Beta the 'star points' occupy some volume, but if the cutoff is the depth where almost all electrons are on an escape path, the the dept from an average radius is modest. At high beta where the average radius is pushed out this is even less significant. The average radius of a roughly spherical volume will give the approximate answer. To refine this to a more accurate number you need to know various parameters like the B field strength, and associated density and energy. Also, the separation of the magnets need to be incorperated, the minor radius of the magnets, etc.

Estimated with a square volume- a simplification to avoid using pi. A cube of 5 liters would be about 17 cm on a side. A quasi sphere would have an average radius a little more than this.

Solving for a sphere of 5 liters volume:

4/3 pi r^3 = 5000cm^3
4.19 r^3= 5000 cm^3
r = cube root of (5000 cm^3/ 4.19)
r (average)= ~10.3 cm

Dan Tibbets

### Re: Help with the math behind the 14 point star

Posted: Wed Jul 22, 2015 2:12 am
A technique I've used to estimate the shape of a plasma wiffleball assuming beta=1:
Take the magrid coils and estimate a set of virtual image coils representing plasma currents. For a polywell the image coil set will look something like a smaller polarity reversed version of the magrid coils. Those points where the image coil field is stronger than the magrid field are inside the plasma.

At a low beta, calculate the magrid field and any inside points where magnetic pressure is below a threshold is inside the plasma. Magnetic pressure is proportionate to magnetic field squared.

Yes, this is computation intensive to calculate every point.

### Re: Help with the math behind the 14 point star

Posted: Thu Jul 23, 2015 3:48 am
Meh,

The University of Sydney solved this one.

Set B equal to a constant and plot the solution. Integrate over X, Y, Z and get the volume.

### Re: Help with the math behind the 14 point star

Posted: Thu Jul 23, 2015 1:53 pm
Is this illustration and formula representative of low Beta or high Beta? At high Beta, the spikes should be very narrow- I think by surface area they represent less than 1/1000th of the total surface area, or perhaps 1/10,000 or by Grads estimate less than 1/100,000 of the contained volume surface area.

And, for what it's worth, I vaguely recall that Dr Parks said that in Mini-B at a Beta of 0.7 the Wiffleball ( weak Wiffle ball border?) was ~ 40-50% of the radius to the magnet mid plane radius. I don't recall precisely.

Dan Tibbets.

### Re: Help with the math behind the 14 point star

Posted: Thu Jul 23, 2015 2:50 pm
To appreciate how narrow the star points are under high Beta conditions look at these images by Indrek.

http://www.mare.ee/indrek/ephi/images.pdf

Dan Tibbets