**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.

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