HP Prime: Black Hole Characteristics – Hawking Radiation

[carey]

(01-05-2018 03:32 AM)Eddie W. Shore Wrote:  HP Prime: Black Hole Characteristics – Hawking Radiation

Equations Used

Given the mass (either in kg or solar masses), the following equations can estimate these black hole characteristics:

Swartzchild Radius (in m):
R = M * G/c^2

Life time left as the black hole slowly radiates (in s):
t = M^3 * 5120 * π * G^2 / (hbar * c^4)

Surface area of the black hole (m^2):
sa = (M^2 * 16 * π * G^2)/c^4

Surface gravity of the black hole (m/s^2), as you can imagine, this will be a huge number:
gr = c^4 / (M * 4 * G)

Universal Gravitation Constant
G = 6.67384 * 10^-11 m^3/(kg * s^2)

Examples

Cygnus X-1: 14.8 solar masses

Schwartzchild Radius: 43705.6410566 m
Life: 6.79261706057 * 10^70 years
Temperature: 4.16932978074 * 10^-9 K (near absolute zero, very cold!)
Surface Area: 24004068275.4 m^2
Surface Gravity: 1.02819127807 * 10^12 m/s^2

While this is a 2018 post, I just noticed it due to the new comments.

Two comments:
1) The code included in the original post is neatly written and motivates me to want to learn Prime programming!
2) An issue regarding precision of the calculated results.

A cardinal result in error propagation is that the result of a calculation can be no more precise than the precision of the least precise number that went into the calculation. Otherwise, precision could be gained by calculation, which it can't, but only by more precise measurements.

The four equations used (Schwarzschild radius, lifetime, surface area, and surface gravity) all depend on G (the universal gravitational constant) which, unfortunately, is known to only 6 digits of precision (while it's value was recently updated, the known precision is still 6 digits). Hence, calculated results using the four equations can only justifiably be expressed to no more than 6 digits of precision, but appear to be expressed to 12 digits of precision.

Actually, the restriction on the number of justifiable digits of precision in the calculated results is even more severe because the four equations also all depend on M. However, M, in the example of Cygnus X-1, is known to only 3 digits of precision (recently updated, but still only 3 digits of precision). Black hole masses (M) are typically known to no more than 3 digits of precision.

Hence, at a minimum, digits 7-12 (digits 4-12 for most examples) of the calculated results would be commonly referred to as "false precision," i.e., while 12-digit results may look more "precise," additional digits of precision beyond the number of digits of precision of the least precise number used in a calculation are meaningless and can detract from this excellent program.