The January 2023 Mathcad Community Challenge was on black holes, because let’s face it, black holes are cool! As a massive star dies, it can end up collapsing upon its own weight. It becomes so dense that it basically punches a hole in the space-time continuum. Here was the challenge:

*Create a worksheet that:*

*Calculates the event horizon (Schwarzschild radius), last photon orbit, last stable particle orbit, and temperature if the sun, moon, and planets of our solar system were to become black holes. (You can choose whether to include Pluto.)**Uses the Chart Component to depict the event horizon and temperature as a function of mass up to the size of UY Scuti, the largest known star in the Milky Way galaxy (which may become a black hole “soon” on a cosmic timescale).*

It turns out that the equations are actually not that complicated. I really wanted people to get a sense of the extraordinary. For example, the Earth has a diameter of 7,917.5 miles (12,756 km). That corresponds to a radius of 3,958.8 miles (6,378 km). If the Earth were a star, it would have to collapse to a Schwarzschild radius of 0.349 inches (8.87 mm). That is nuts!

I also asked for temperature because I was surprised that black holes have a temperature. They are so massive that not even light can escape a black hole, but Professor Stephen Hawking figured out that black holes emit radiation and therefore have a temperature. A very, very low temperature, but a temperature, nonetheless. Absolute zero is -459.67 degrees Fahrenheit (-273.15 degrees Celsius). If the Earth were a black hole, its temperature would be -459.633 degrees Fahrenheit (-273.129 degrees Celsius).

The first entry was made by user PGrist. I had provided a NASA reference for the equations, but I mistakenly directed people towards the equation that calculates Schwarzschild radius as a function of the mass of the sun. I prefer the “purer” equation that calculates the event horizon as a function of object mass. PGrist calculated the event horizon, last photon orbit, and last stable particle orbit, but left the values in kilometers. Expressing the values between 10 to the -7 power and 10 to the -10 power diminishes the impact of the results.

The second submission was by frequent contributor Fred Kohlhepp. He created some nice XY plots depicting event horizon radius and temperature as a function of mass. Fred also used a table to compile the masses of the sun, moon, and planets. By vectorizing the mass column, Fred placed a second trace for the solar system objects onto the plot. Nice.

The third and final submission was by another frequent contributor, PPal. Like PGrist, PPal calculated the temperatures at the event horizon, last photon orbit, and last stable particle orbit. (It turns out there are multiple temperature calculations for black holes.) PPal created nice XY Plots as well. I like the plotting of the names of celestial objects on the X-axis in some plots as well as logarithmic scales on both axes in others.

I was disappointed that none of the submissions depicted the Schwarzschild radius or temperature using the Chart Component. Everyone used the old XY 2D plot. I understand that people might be more familiar with the standard XY plot, and it has functionality you can’t find elsewhere. But for creating publication-ready images, the Chart Component is the way to go. Here is an example of the Chart Component I made depicting the event horizon and temperature as a function of mass:

There’s an old saying that math is the language of physics. Mathcad’s documentation and graphing tools can help you convey the real meaning behind phenomena and its meaning. Join us in March for an electrical engineering related challenge!

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