Cosmic velocities are the velocities that a body must reach at least in order to leave a celestial body on an orbit around this celestial body or around this celestial body.
A distinction is made between three different cosmic speeds:
- the 1st cosmic speed (minimum orbital speed)
- the 2nd cosmic speed (escape speed from the earth)
- the 3rd cosmic velocity (escape velocity from the solar system)
These three cosmic speeds are essential for space travel. With their help, it is possible to estimate the final speed rockets must have in order to bring a satellite into a stable orbit or to transport people to other celestial bodies or to leave our solar system with a probe.
The first cosmic speed – orbital speed
In order to remain in a stable orbit around the earth, each satellite needs a certain speed. He achieves this in the first few minutes after the start through the thrust of the carrier rocket. A rough estimate of this speed is called the first cosmic speed or also the minimum orbital speed.
The first cosmic velocity thus describes how fast a body would have to be launched horizontally from the earth’s surface in order to remain powerless on the earth’s circular path without falling back to the earth’s surface. In practice, however, this is not possible because of the high air resistance and the mountains on earth.
Calculation of the first cosmic velocity
For the earth, the value of the first cosmic velocity is \({v}_1\) = \(\text{7.91} \frac{km}{s}\). The orbital velocity can also be calculated for other celestial bodies. The formula for this is as follows:
\begin{align} v_1 &= \sqrt {G \cdot \frac {M}{r}} \end{align}
\( G = 6.67 \cdot 10^{-11} \frac{m^3}{kg \cdot s^2} \)
\( M = 5.97 \cdot 10^{24} \, kg \)
\( r = 6,371 \, km \)
G stands for the gravitational constant. If you want to learn more about them, go to the article on Newton’s law of gravitation.
The M describes the mass and r is the radius of the celestial body.
When was the first cosmic speed needed for the first time?
The first satellite was launched into stable orbit around the Earth on October 4, 1957. As we have already learned, the calculation of the first cosmic velocity is relevant for this. The first artificial satellite is called Sputnik 1 and was sent into space by the then Soviet Union, from where it sent radio signals to Earth.
- The first cosmic speed is also called circular speed.
- It describes how fast a body has to be in order to get into a stable orbit around the earth.
- For the earth the first cosmic velocity is v= 7.910 km/s
- For the calculation of the first cosmic velocity the formula is v= square root of G • M/r
- Sputnik 1 was the first satellite sent into space.
The second cosmic velocity – escape velocity from the earth
For example, in order to land on the moon, a space shuttle needs such a high speed that it can move away from the gravitational area of the earth or any other celestial body. The second cosmic speed is a rough estimate of this speed. It is therefore also called the escape velocity from the earth.
\begin{align} E_{kin}&=F_G \\\frac{1}{2}\cdot m\cdot v_{2}^{2}&=G\cdot\frac{M\cdot m}{r }\\v_2&=\sqrt{2\cdot\frac{G\cdot M}{r}}\text{,}\end{align}
The second cosmic velocity thus describes how quickly a body would have to be launched from the earth in order to leave the earth’s gravitational field without power. Calculating the speed is important, for example, if you want to send space probes from Earth to the Moon or Mars. However, this does not take into account the Earth’s own rotation and swing-by maneuvers on other planets.
A swing-by maneuver designates a method of space travel. A light missile, such as a probe, flies close to a very heavy body, such as a planet. The direction of flight of the probe is changed and the speed increases or decreases.
Calculation of the second cosmic velocity
For the earth, the second cosmic speed has the value v= 11.19 kilometers per second.
But this speed can also be calculated for other celestial bodies. The formula is:
v= square root of 2G • M/r
The letters in this formula stand for the same magnitudes as in the formula for the first cosmic velocity.
When was the second cosmic speed needed?
On July 21, 1969, Neil Armstrong became the first man to walk on the moon. The Apollo 11 mission brought a human being to another celestial body for the first time that day, for which the second cosmic speed was required.
- The second cosmic velocity is also called the escape velocity from the earth.
- It describes how fast a body has to be to leave the earth’s gravitational pull.
- The Earth’s own rotation and swing-by maneuvers on other planets are ignored in the calculation.
- For the earth the second cosmic velocity is v= 11.19 km/s
- The formula for the calculation is: v= square root of 2G • M/r
- Neil Armstrong was the first person to set foot on another celestial body.
The third cosmic velocity – escape orbit velocity from the solar system
If we want a probe to escape the sphere of attraction between the Earth and the Sun and leave our solar system, it needs a very high speed. A rough estimate of this speed is called the third cosmic speed. It is also called the escape orbit velocity from the solar system.
The third cosmic speed describes how fast a body has to be launched from the surface of the earth in order to leave the gravitational field of the earth and the sun without power. Here, too, the earth’s own rotation or swing-by maneuvers on other planets are not taken into account. However, the orbital speed of the earth around the sun is taken into account.
Calculation of the third cosmic velocity
The third cosmic speed cannot be calculated exactly. When considering such a size, many factors play a role, not all of which can be taken into account in the calculation. The sun, the earth, and the probe or body under consideration form a three-body problem that cannot be solved exactly.
Nevertheless, an approximate value of v~ 16.67 kilometers per second was calculated for the earth.
To calculate the third cosmic velocity of other celestial bodies, the following formula is used:
v~ square root of 2G • M of the sun / r of the earth
When was the third cosmic speed needed?
In 2012 it was reported that Voyager 1 had reached the edge of the heliosphere. This makes it the most distant probe from Earth and the Sun that has been launched by humans. It is expected to leave the Sun’s gravitational hold in about 56,000 years.
- The third cosmic velocity is also called the escape velocity from the solar system.
- So it describes how fast a body has to be in order to escape the gravitational pull of the earth and the sun.
- The calculation does not take into account the Earth’s own rotation or swing-by maneuvers on other planets, but the Earth’s orbital speed around the sun.
- For the earth the value of the third cosmic speed v~ is 16.67 km/s.
- The third cosmic speed is calculated with the formula v~ square root of 2G times M of the sun / r of the earth
- It is estimated that Voyager 1 will leave the Sun’s gravitational hold in 56,000 years.
Everything you need to know about cosmic speeds at a glance!
- The first cosmic speed describes how fast a body has to go in order to get into a stable orbit around the earth. For the earth it is v= 7.910 km/s
- For the calculation of the first cosmic velocity the formula is v= square root of G times M/r
- The second cosmic speed describes how fast a body has to be in order to leave the earth’s area of attraction. For the earth it is v= 11.19 km/s.
- The formula for calculating the second cosmic velocity is v= square root of 2G times M/r
- The third cosmic speed describes how fast a body has to be in order to escape the attraction of the earth and the sun. For Earth, its value is approximately v~ 16.67 km/s.
- The third cosmic speed is calculated with the formula v~ square root of 2G times M of the sun / r of the earth
proof
- Spektrum.de: Cosmic speeds (08/11/2022)
- Academic.com: Cosmic Speed (08/11/2022)
- Hu-berlin.de: Free fall around the earth (08/11/2022)