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Space object velocity

It calculates the circular orbital velocity, escape velocity and planetary system escape velocity for the given planet parameters.
Anton2017-04-12 18:34:40

Since ancient times, people tried to imagine how our world works.
Aristarchus of Samos was ancient Greek philosopher. In early III b.c. he suggested a heliocentric world model (the Sun in the center). He also tried to calculate sizes of Earth and Sun and distance between them using Moon position. There were many opposers standing for geocentric system (the Earth in the center), so the heliocentric idea did not get much follow up.
It took almost two thousand years to recover it. Nicolaus Copernicus, the Polish astronomer reformulated the model of universe with Sun in the center. The work of whole his life was published in 1543, the year of his death. Copernicus heliocentric model and tables of planetary positions more accurately reflected the observed state.
Half century later in 1609, Johannes Kepler, German mathematician published planetary motion laws, which improved accuracy of Copernicus model. Kepler created his laws as result of analysis of big amount of data,carefully collected by Tycho Brahe, Danish astronomer.
The end of 17th century was marked by the great English scientist, Isaac Newton discoveries. Newton's motion laws and universal gravitation law provided the theory basis and extended Kepler formulas.
Finally in 1921, Albert Einstein issued the general theory of relativity (GTR), which describes gravity phenomena and planetary motion mechanics with highest accuracy. In most cases relativistic effects can be ignored and Newton classical laws give a fairly accurate planet motion description. So thanks to Newton and his predecessors, now we still can calculate:

  • the circular orbital velocity of a satellite object (first space velocity)
  • the speed of object to escape of planet gravitation (second space velocity)
  • the speed of object to escape planetary system gravitation (third space velocity)
    using very simple formulas.

Space object velocityCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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Circular orbital velocity

Circular orbital velocity , is the speed required to keep circular object motion at specified altitude above the planet.
The equation is: v_1=\sqrt{{\frac{GM}{R}}},where
R=r+h - orbit radius, combined by r -planet radius and h - altitude above the planet
M - planet mass
G - gravitational constant 6.67408(31)10-11 m³/(s²·kg)
The formula can be easily derived from Newton's gravitation force formula and centrifugal force formula:
m\frac{v_1^2}{R}=m\frac{GM}{R^2}
m -object mass (excluded, during evaluation of v1)

Two and half centuries later after Newton discoveries, in 1957, USSR launched the first artificial satellite of the Earth. R-7 carrier rocket overcame the atmosphere resistance and the Earth gravity to deliver Sputnik-1 to the 577-km orbit.

Escape velocity

is a speed required to escape gravitational influence of the planet or star.
The formula is: v_2=\sqrt{{\frac{2GM}{R}}}
It correlates with v1 as follows: v_2=\sqrt{2}{v_1}
The formula can be derived from the kinetic energy and Mechanical work done to overcome the gravity moving object from the initial altitude to infinity:
m\frac{v_2^2}{2}=\int \limits _{R}^{\infty}{m\frac{GM}{r^2}{dr}=-\frac{mGM}{r}\Big{|}_{\small{R}}^{\infty}=m\frac{GM}{R}

In 1959 USSR launched Luna-1, the automatic interplanetary module, which overcame influence of the Earth gravity and becomes 1st artificial satellite of the Sun.

Planetary system escape velocity

is a minimal speed required to overcome whole planetary system gravity including planet and star gravity.
v_3=\sqrt{(\sqrt{2}-1)^{2}v^{2}+v_2^2},
where v - planet orbital speed
v2 - planet escape speed
Voyager-1 record for alience

According to calculations, the unit launched from the Earth must have velocity of 16.6 km/s, to leave Solar system.
Nearest velocity (16.26 km/s) was achived by "New Horisons" module launched in 2006 by USA to research the Pluto and Charon, its satellite. Up to date, the module finished Pluto filming, it goes toward Kuiper belt.
The first world module, which achive Solar system escape speed was "Voyager-1", launched by Unated States in 1997. Initial "Voiyager" speed was lower than "New Horisons" module speed, but due to series of gravitational maneuvers against Solar system planets, especially Jupiter, the speed raised up to 17 km/s. Currenly, the "Voiyager-1" left Solar system. It still collect and send data to the Earth. The unit carries 12-inch gold plate copper disc containing audio-visual records form the Earth intended for intelligent life forms from other planetary systems.


Sources:
V. Zakharov, The gravitation: from Aristotle to Einstein
Photo of Voyager Record, NASA, Voyager Project.

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