Orbital Velocity Calculator (First Cosmic Velocity)

Definition of the first cosmic velocity

The first cosmic velocity is the speed at which a body in the gravitational field of a mass M closes a circular orbit of radius r and needs no further thrust to stay on it. Gravity is then the only centripetal force. The total energy on that orbit is negative and equal to −GMm/(2r): in the sense of the virial theorem, half of the potential energy has become kinetic energy. A slower body falls on an ellipse or a straight line. A faster body, up to v₂, is still bound, but the path becomes an ellipse with one focus at the center of mass.

Three properties follow. First, v₁ does not depend on the satellite’s mass. Second, the velocity is tangent to the path: the radial component is zero, or the radius would grow or shrink. Third, v₁ falls as 1/√r and the period grows as r to the power 3/2. A geostationary satellite is therefore slower than the ISS, even though it orbits much higher, and its period matches a sidereal day.

From Kepler to Sputnik

Johannes Kepler, in Harmonices mundi (1619), wrote that the square of the orbital period is proportional to the cube of the mean distance from the Sun. Isaac Newton showed in the Principia (1687) that this proportion follows from a force that falls as 1/r². The thought experiment of a shot from a high mountain, described in De mundi systemate (1728), separates two thresholds: the speed at which the cannonball circles the Earth, and the higher speed at which it never returns. The name “first cosmic velocity” is the school term in Polish and Russian; English textbooks usually say circular orbital speed.

The first artificial satellite of Earth was Sputnik 1, launched on October 4, 1957, into low Earth orbit. The speed of such an orbit is closer to the 7.67 km/s of the ISS example than to the textbook 7.91 km/s computed at the surface itself. The surface result remains a reference, not a flight plan.

Derivation from the centripetal force

Step 1. The gravitational force on a satellite of mass m is F = GMm/r². Step 2. On a circle that same force is the centripetal force mv₁²/r, so GMm/r² = mv₁²/r. Step 3. Cancel m (m > 0) and multiply both sides by r: v₁² = GM/r, so v₁ = √(GM/r). Step 4. For Earth, with G = 6.67430·10⁻¹¹ N·m²/kg², M = 5.972·10²⁴ kg, and r = 6.371·10⁶ m, GM = 3.986·10¹⁴ m³/s² and v₁ = 7,910 m/s.

The period comes from the circumference: T = 2πr/v₁ = 2π√(r³/GM). At Earth’s surface that is 84.3 min. Substituting r = 42,164 km gives 23.9 h, the sidereal day of a geostationary orbit in this model. Written through g = GM/r², the same formula is v₁ = √(gr). At Earth’s surface, √(9.82·6.371·10⁶) returns 7.91 km/s.

Escape speed and the ellipse

The second cosmic velocity, v₂ = √(2GM/r), comes from energy, not from the centripetal force, and it is exactly √2 times v₁. For Earth, 7.91 km/s·√2 = 11.19 km/s. The kinetic energy of escape is twice the kinetic energy of the circular orbit, because the speed rises only by 41 percent while energy rises with the square of speed. Between v₁ and v₂ the path is an ellipse. Speed on an ellipse follows the vis-viva equation v = √[GM(2/r − 1/a)], which this calculator does not evaluate.

The third cosmic velocity, about 16.7 km/s relative to Earth, is enough to leave the Solar System. It is not the sum of v₁ and Earth’s speed around the Sun. The calculator does report Earth’s circular speed at 1 AU (29.79 km/s) as an example that uses the solar mass, without adding vectors.

Assumptions and what the model leaves out

By Newton’s shell theorem, the field outside a spherically symmetric body is the field of a point with the same mass. Earth is oblate, so a real GM and real orbits need geopotential harmonics; the school square root does not see them. The satellite’s mass must be much smaller than M. For two comparable bodies the numerator is M + m.

The model ignores air. At the altitude of the ISS drag is small, but it is enough that the station loses altitude without corrections. An eastward launch from the equator adds about 0.465 km/s relative to the ground, yet v₁ in this calculator is the speed relative to the inertial center of mass, not relative to the launch site. A rocket does not reach orbit with a finished v₁ at the ground: it accelerates gradually and meets the speed that belongs to its r only after it is above the dense atmosphere.

Sources

The definition, the derivation, and the numbers come from mechanics textbooks and metrology sources: I. Newton, Philosophiae Naturalis Principia Mathematica, 1687, and De mundi systemate, 1728; J. Kepler, Harmonices mundi, 1619; D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, gravitation; H. D. Young and R. A. Freedman, University Physics, “Gravitation”; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 7; H. D. Curtis, Orbital Mechanics for Engineering Students, Butterworth-Heinemann; CODATA 2018 (NIST) for G; NASA Planetary Fact Sheets. This page is educational.