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Mathcad Community Challenge January 2023 - Black Holes

DaveMartin
16-Pearl

Mathcad Community Challenge January 2023 - Black Holes

This month’s challenge is on black holes! 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).

Black_hole's_accretion_disk.jpg

 

Image courtesy of NASA Goddard Space Flight Center / Jeremy Schnittman used under CC BY-SA 4.0.

 

Considerations:

  • How do you provide the inputs? Matrix, table, combo box input control, or other?
  • When creating the Chart Component, does it make sense to use a log scale anywhere?
  • The equations can be found here (hint: most are on page 36).
  • Documentation (text blocks, text boxes, and images) for this challenge is critical! The math is actually not that complicated so presentation matters. Someone should be able to understand the worksheet without someone explaining it to them.

Are you up to the challenge?

Be sure to consult the PTC Mathcad Community Challenge Guidelines.

 

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com
7 REPLIES 7
PGrist
4-Participant
(To:DaveMartin)

Thought I'd have a crack at this one. Note this is MathCAD 7 format.

ppal
17-Peridot
(To:PGrist)

I have a question.

 

From the reference document

 

Problem 1 - Sketch a life-sized illustration of the gas surrounding the above black hole
and give the temperature at a distance of 1 meter, 50 centimeters and 5 centimeters from
the center of the black hole.
Answer: At 1 meter, T = 35,000 K, which is 6 times the surface temperature of our sun.
At 50 cm or 0.5 meters, T = 59,000 K.
At 5 centimeters or 0.05 meters, T = 331,000 K

 

Using your equation i get a different answer

ppal_1-1672954963075.png

Your thoughts on my equation. Thanks.

 

A couple tips:

  • Take notice of the units you are going to use to express the Schwarzschild radius. Maybe it should be something small. The person reading the worksheet should have the effect of, "Holy cow, if I shrunk the Earth / Moon / Jupiter to this small size, it would collapse into a black hole!"
  • A simple equation for temperature based on mass can be found about halfway down on the following link. I included temperature in the challenge because I was shocked to find that black holes have a temperature! I thought they absorbed everything including heat. Stephen Hawking is (was) simply brilliant.

https://www.omnicalculator.com/physics/black-hole-temperature  

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com
PGrist
4-Participant
(To:ppal)

Aha! I see you've found my deliberate error 😉

Thank you for the peer review.

Updated version attached (also added the log scaling to the temperature graph).

Not terribly challenging except for having to mine equations from the reference.  Note that there is a difference between the "gas temperature" and the black body temperature equations.  Prime 4 Express

January 2023 only lasts for another week!

 

If you want to submit something that'll have an impact on the challenge discussion, please submit a worksheet using the chart component!

I manage the Creo and PTC Mathcad YouTube channels for PTC, as well as all PTC Mathcad marketing in general.
ppal
17-Peridot
(To:DaveMartin)

In Prime 8

 

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