01-05-2018, 03:32 AM
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
(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
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