HP Prime: Black Hole Characteristics – Hawking Radiation

[Eddie W. Shore]

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)

The HP Prime uses the conversion for seconds to years:
1 year ≈ 31556925.9747 s

Average temperature of the black hole (K):
temp = (hbar * c^3)/(M * 8 * k * π * G)

If you need to convert to °C subtract 273.15 from this result.

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)

Constants Used

Values are taken from the HP Prime

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

Speed of Light
c = 299792458 m/s

Plank Constant Divided by 2*π
hbar = 1.054571726 * 10^-34 J*s

Boltzmann Constant
k = 1.3806488 * 10^-23 J/K



HP Prime Program BLACKHOLE

Code:

EXPORT BLACKHOLE()
BEGIN
// EWS 2018-01-04
// Hawking Radiation

// Input of mass
LOCAL ch,M;
INPUT({M,{ch,{"kg","Solar Mass"}}},
"Mass",{"Mass: ","Unit: "});

IF ch=2 THEN
M:=1.988435ᴇ30*M;

END;

LOCAL k,G,c,hbar;
// from 4 Constants
// Shift, Units, Constant
k:=1.3806488ᴇ−23; // Chemistry
G:=6.67384ᴇ−11; // Physics
c:=299792458; // Physics
hbar:=1.054571726ᴇ−34; // Quantum

// Swartzchild radius (m)
LOCAL R:=(2*G*M)/c^2;

// Life (years)
LOCAL t:=M^3*5120*π*G^2/
(hbar*c^4*31556925.9747);

// Temperature (K)
LOCAL temp:=(hbar*c^3)/
(M*8*k*π*G);

// Surface Area (m^2)
LOCAL sa:=M^2*16*π*G^2/c^4;

// Gravity 
LOCAL gr:=c^4/(4*G*M);

// Results
PRINT();
PRINT("Schwartzchild Radius:");
PRINT(R+" m");
PRINT("Life:");
PRINT(t+" years");
PRINT("Temperature:");
PRINT(temp+" K");
PRINT("Surface Area:");
PRINT(sa+" m^2");
PRINT("Surface Gravity:");
PRINT(gr+" m/s^2");


// List
RETURN {R,t,temp,sa,gr};
END;

(edited 1/23/2018, see thread below)

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

Sagittarius A*, the center of our Milky Way Galaxy: 4.31 million solar masses

Schwartzchild Radius: 12727791415.8 m
Life: 1.67758214773 * 10^87 years
Temperature: 1.43169560917 * 10^-14 K
Surface Area: 2.03571024785 * 10^21 m^2
Surface Gravity: 3530680.02679 m/s^2

Source:

Jim Wisniewski. “Hawking Radition Calcualtor” Xaonon. January 3, 2017. http://xaonon.dyndns.org/hawking/ Retrieved December 26, 2017

[salvomic]

Thanks!
Very useful program.

Salvo

[hfw]

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

<...>
HP Prime Program BLACKHOLE

Code:

EXPORT BLACKHOLE()
BEGIN
// EWS 2018-01-04
// Hawking Radiation

// Input of mass
LOCAL ch,M;
INPUT({M,{ch,{"kg","Solar Mass"}}},
"Mass",{"Mass: ","Unit: "});

IF ch:=2 THEN
M:=1.988435ᴇ30*M;

<...>

<...>
http://xaonon.dyndns.org/hawking/ Retrieved December 26, 2017

Hello,

nice program - had to try it immediately with my brand new HP Prime

Yet - it has a small error: If you try to change the input units to [kg], instead of [solar masses], the the check goes wrong:

Code:

IF ch:=2 THEN

should read

Code:

IF ch=2 THEN

With this small change it works nicely and delivers good results. The bug is in IF ch:=2, this sets ch to 2, instead of comparing it.

[Eddie W. Shore]

I think the line should be

IF ch==2 THEN

since two equal signs are used for comparison. It seems to work either way with the one or two equal signs. The code on the original thread is now corrected.

Much appreciation!

Eddie

[morvaoutig]

Bonjour,
Je prends connaissance de ce très intéressant programme ASTROLAB pour HP PRIME que je vais proposer à mon petit fils.
Quelqu'un pourrait-il me renseigner du nom de l'auteur? de ses coordonnées?
J'ai pris connaissance du "Guide" mais ne trouve pas le nom de ce génial astronome.
Merci d'avance.

[rprosperi]

Please use English, even if just a simple online translation, so most readers can understand. Fee free to also include your original language as well, as often there are subtle nuances that are better expressed in the native language.

For your post, google provided this, which seems clear:

"I am learning about this very interesting ASTROLAB program for HP PRIME that I am going to offer to my grandson.
Can anyone tell me the name of the author? his contact details?
I read the “Guide” but cannot find the name of this brilliant astronomer."

[morvaoutig]

Good morning,
I am learning about this very interesting ASTROLAB program for HP PRIME which I am going to offer to my grandson.
Can anyone tell me the name of the author? his contact details?
I read the “Guide” but cannot find the name of this brilliant astronomer.
Thanks in advance

[KeithB]

(01-05-2018 09:05 AM)salvomic Wrote:  Thanks!
Very useful program.

Salvo

Wow, what do you do for a living to make this "useful"? 8^)

[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.