RPN: HP 11C: Surface Gravity and Escape Velocity
EQUATIONS
The surface gravity constant of a celestial object (planet, dwarf planet, star, etc.):
g_p = G * M ÷ R²
The escape velocity of a celestial object:
v_esc = √(2 * G * M ÷ R)
where (using SI units):
g_p: surface gravity (m/s)
v_esc: escape velocity (m/s)
M: (measured) mass of the object (kg)
R: (average) radius of the object (m)
G: Universal Gravitational Constant (G ≈ 6.6743 * 10^-11 N m²/kg² (or m³/(s² kg))
The value of G is the 2022 CODATA value (https://physics.nist.gov/cgi-bin/cuu/Value?bg)
Determining Surface Gravity and Escape Velocity
DERIVATION - Determine the surface gravity constant in terms of escape velocity.
Start with the escape velocity:
v_esc = √(2 * G * M ÷ R)
(v_esc)² = 2 * G * M ÷ R
dividing both sides by 2 (we'll see why this important in a bit):
(v_esc)² ÷ 2 = G * M ÷ R
(v_esc)² * 1/2 = G * M * 1/R
Then insert the square of escape velocity in the equation for the surface velocity:
g_p = G * M ÷ R²
g_p = G * M * 1/R²
g_p = G * M * 1/R * 1/R
g_p = (v_esc)² * 1/2 * 1/R
g_p = (v_esc)² ÷ (2 * R)
The equations will the be:
v_esc = √(2 * G * M ÷ R)
g_p = (v_esc)² ÷ (2 * R)
Set the stack up as:
Y: M (mass, kg)
X: R (radius, m)
The results are shown in the stack:
Y: g_p (surface gravity, m/s²)
X: v_esc (escape velocity, m/s)
Algorithm (done with an HP 11C):
ENTER
ENTER
R↑
2
×
6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)
×
R↑
÷
ENTER
√ (view escape velocity)
R↓
x<>y
÷
2
÷ (view surface gravity)
R↑ (set surface gravity in the Y stack, escape velocity in the X stack)
Example:
Estimate the surface gravity constant and escape velocity of Venus.
Venus
Mass ≈ 4.8675 * 10^24 kg
Radius ≈ 6.0518 * 10^6 m
Surface gravity ≈ 8.8704 m/s
Escape velocity ≈ 10361.6414 m/s
Determining a Planet's Radius and Escape Velocity
Problem: Given Earth's surface gravity is defined as 9.80665 m/s and mass of 5.972168 * 10^24. Estimate the radius and escape velocity.
Here we are given g_p and M, and we are tasked with finding R and v_esc.
Start by solving for R:
g_p = G * M ÷ R²
Multiply by R² and divide by g_p. Keep this in mind.
R² = G * M ÷ g_p
Take the square root and solve for the radius.
R = √(G * M ÷ g_p)
Note that:
R² = G * M ÷ g_p
g_p * R² = G * M
2 * g_p * R² = 2 * G * M
2 * g_p * R = 2 * G * M ÷ R
This makes for an easy substitution for v_esc.
v_esc = √(2 * G * M ÷ R) = √(2 * g_p * R)
The equations used are:
R = √(G * M ÷ g_p)
v_esc = √(2 * g_p * R)
The algorithm uses one memory register, I just picked R0 (done with an HP 11C):
STO 0
÷
6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)
×
√ (view R)
ENTER
RCL 0
×
2
×
√ (view v_esc)
Set the stack up as:
Y: M (mass, kg)
X: g_p (surface gravity, m/s²)
The results are shown in the stack:
Y: R (radius, m)
X: v_esc (escape velocity, m/s)
Results:
Inputs:
Mass of Earth ≈ 5.972168 * 10^24 kg (enter as the y stack)
Surface Gravity = 9.80665 m/s² (enter as a x stack, and yes, surface gravity of Earth is defined to be exactly 9.80665 m/s²)
Outputs:
Y: Radius of Earth ≈ 6375416.060 m
X: Escape Velocity ≈ 11182.2604 m/s
Sources
The NIST Reference on Constants, Units, and Uncertainty. "Newtonian constant of gravitation" Fundamental Physical Constants. Last updated May 9, 2024. https://physics.nist.gov/cgi-bin/cuu/Value?bg Retrieved September 4, 2025.
Research & Education Association. The Essentials of Astronomy Piscataway, New Jersey. 2004. ISBN 0-87891-965-1
Eddie
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