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
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;
Note: (1/23/2018)
The line in red used to read IF ch:=2 THEN, this is an error on my part. The above code is now correct. (Alternatively, IF ch=2 THEN also works) Much appreciation to hfw and Salvomic on HP Museum of Calculators for pointing this error to me!
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
Eddie
This
blog is property of Edward Shore, 2018.