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 HP Prime: Black Hole Characteristics – Hawking Radiation
01-05-2018, 03:32 AM (This post was last modified: 01-23-2018 03:02 PM by Eddie W. Shore.)
Post: #1
 Eddie W. Shore Senior Member Posts: 1,596 Joined: Dec 2013
HP Prime: Black Hole Characteristics – Hawking Radiation
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:

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;

Examples

Cygnus X-1: 14.8 solar masses

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

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
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 Messages In This Thread HP Prime: Black Hole Characteristics – Hawking Radiation - Eddie W. Shore - 01-05-2018 03:32 AM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - salvomic - 01-05-2018, 09:05 AM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - KeithB - 03-12-2024, 02:42 PM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - hfw - 01-20-2018, 12:23 PM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - Eddie W. Shore - 01-23-2018, 03:01 PM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - morvaoutig - 03-12-2024, 10:55 AM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - rprosperi - 03-12-2024, 12:08 PM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - morvaoutig - 03-12-2024, 12:50 PM RE: HP Prime: Black Hole Characteristics – Hawking Radiation - carey - 03-12-2024, 03:22 PM

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