Normality Calculator for Standard Acids

Calculating Escape and Circular Velocity with Python

In orbital mechanics, there is a beautiful mathematical harmony between the velocity required to maintain a circular orbit and the velocity required to escape a planet's gravity entirely. This relationship is defined by a factor of the square root of 2.

The Formula:
The escape velocity ($v_{esc}$) is related to the circular orbital velocity ($v_{circ}$) as follows:
v_esc = v_circ * sqrt(2)

The following Python script calculates these values for different altitudes, including the Earth's surface, the summit of Mt. Everest, and Geostationary Orbit (GEO).

import math

def escape_vel(d):

    # Using the constant 894 for Earth calculations

    # d is distance from Earth's center in km

    return 894 / math.sqrt(d)

# Radius of Earth in km

rad_earth = 6378

# Locations and their altitude from surface (km)

distances = {

    "Surface Earth": 0,

    "Everest": 8.848,

    "ISS Orbit": 408,

    "GEO Orbit": 35786

}

print(f"{'Location':<15} {'Alt (km)':<10} {'Esc Vel':<12} {'Circ Vel'}")

print("-" * 50)

for name, alt in distances.items():

    r = rad_earth + alt

    ve = escape_vel(r)

    vc = ve / math.sqrt(2)

    print(f"{name:<15} {alt:<10.1f} {ve:<12.4f} {vc:<12.4f}")

    

Key Takeaways

• Earth's Surface: Escape velocity is ~11.18 km/s.
• Low Earth Orbit (LEO): At the altitude of the ISS, the velocity required to stay in orbit is roughly 7.67 km/s.
• GEO: At high altitudes, gravity weakens, significantly reducing the velocity needed to maintain orbit.