Shulou(Shulou.com)11/24 Report--

Author: Cong Yu

Audit: time

The full text is about 2500 words, and it takes 10 to 15 minutes to read.

Introduction how many steps does it take to send the probe to Mars? Generally speaking, the flight process of a probe from Earth to other planets is roughly divided into three stages: a rocket is launched and orbited around the Earth, accelerated to escape gravity to the target planet, and upon arrival, it orbits or lands on the target planet. The first and third stages are short but complex in steps and details, while the second phase will take up most of the aircraft's journey. Starting from the motion and orbit problems of spacecraft, this paper will introduce some simple knowledge of celestial motion and orbit transfer.

The three cosmic speeds we will start with the problem of the speed of the spacecraft.

As we all know, the faster the initial velocity of an object thrown in the horizontal direction is, the farther the landing point is. When its speed reaches 7.9 km / s, it will never fall back to the ground and will move in a uniform circle around the earth only under the influence of the earth's gravity. This is the first cosmic speed of the earth, the minimum initial velocity for launching a satellite, and the maximum linear velocity for orbiting the earth in a circular orbit.

If the initial speed of launch reaches about 11.2 km / s, that is, the second cosmic speed of the earth, the vehicle will be able to escape the earth's gravity completely, and there is no need to accelerate afterwards. The second cosmic velocity of a celestial body is also called escape velocity. An unpowered object emitted from the surface of a celestial body in any direction with an escape velocity decreases slowly under the action of gravity and happens to be zero when flying to infinity. The escape velocity is times the first cosmic velocity, which can be calculated by the mutual conversion of kinetic energy and gravitational potential energy (zero point of potential energy at infinity). In this process, kinetic energy is gradually converted into gravitational potential energy until both are zero at infinity.

The third cosmic velocity is the minimum initial launch velocity needed to break away from the gravity of the earth and the sun at the same time, which is different from the second cosmic velocity in any direction. It requires that the initial motion direction of the aircraft must be the same as that of the earth, so that the fuel consumption can be reduced with the help of the speed of the earth. According to the above, the escape velocity of the sun in the earth's orbit is 29.8 km / s, about 42.1 km / s, so the aircraft not only needs the kinetic energy corresponding to 12.3 km / s to escape from the solar gravity, but also needs the kinetic energy corresponding to 11.2 km / s to escape from the earth. The sum of these two kinetic energy values corresponds to the third cosmic speed 16.7 km / s.

Obviously, interplanetary vehicles commuting between the earth and other planets, the destination is far from the earth, it can be said that the need to completely overcome the gravity of the earth, but will not leave the gravity range of the sun, so their speed is usually between and.

The derivation process of the second and third cosmic velocities secondly, we consider the specific orbital shapes corresponding to different orbital velocities and launch directions.

In the simplest case, the object is thrown horizontally. The orbit shape of the first cosmic velocity is circular, and if the velocity is slightly greater than this, the centripetal acceleration of the object will be greater than that provided by gravity, and the orbit will be an ellipse-focusing on the center of the earth and launching at perigee. With the increase of the initial velocity, the ellipse will become more and more flat, and the apogee height, semi-major axis and eccentricity will gradually increase. If launched at the second cosmic velocity, the eccentricity of the orbit will become 1, and the distance between the semi-major axis and apogee is infinite, that is, an open parabolic orbit. If the initial velocity ratio is still large, the orbit of the aircraft relative to the earth will be a hyperbola, the eccentricity is greater than 1, and the semi-major axis is defined as a negative value in celestial mechanics.

Looking back at those flat projectile motions whose launch velocity is less than, strictly speaking, their trajectories are also ellipses, but the initial velocity is very small, the ellipse is also very flat, and the eccentricity is infinitely close to 1, so it can be regarded as a parabola. At this time, the initial ejection position is the apogee of the ellipse and the perigee is in the interior of the earth.

In fact, except for the special case of vertical upward emission, when the initial velocity of an object is less than, equal to, and greater than, even if the launch direction is not horizontal, the orbit will still be ellipse, parabola and hyperbola respectively, but the semi-major axis, eccentricity and focal length of parabola are also different in different directions. For cases where these directions are not horizontal, if the velocity and vector diameter length are known, we can judge the shape of the orbit and the size of the semi-major axis by calculating the mechanical energy of the object. the sum of kinetic energy and potential energy less than, equal to or greater than 0 corresponds to elliptical, parabolic and hyperbolic orbits, respectively. Because of the mechanical energy of the system in the two-body problem, their semi-major axes can be obtained.

Orbit change motion and Horman transfer when V1 and v2 are launched in different directions. Finally, let's come to the topic of spacecraft orbit change. As the actual orbit transfer and track design must be extremely complex, so the following only introduce some simple theories and ideas.

Consider a spacecraft or satellite orbiting the earth in low earth orbit that suddenly ignites and accelerates in its original direction of motion. At this time, the velocity of its motion will be greater than that of the circular orbit here, the centripetal acceleration will also be greater than that needed for circular motion, and the orbit will become an elliptical orbit with the acceleration position as perigee, which is the same as the case where the surface velocity is between and in the case we discussed above. When it reaches the apogee, the spacecraft makes another appropriate acceleration along the direction of motion, so that the speed is exactly equal to the orbiting speed of the current distance. it then successfully changed its orbit to a circular orbit with a larger radius and a higher height than the original-- this is the basic idea of the orbit change operation. To reduce the orbit height, you only need to reverse the operation, slow down to the elliptical orbit, and decelerate again at the perigee.

The diagram of the orbit change process of the satellite, of course, the orbit after the orbit change does not have to be round, and the energy released by a single ignition of the engine may not be enough to reach the target height in a short period of time. It can be ignited again after the satellite returns to the initial position around a circle, or it can be accelerated many times every time it reaches the apogee. At this time, the satellite orbit is a set of elliptical rings of different sizes. The position of acceleration (or deceleration) is not limited to apogee (or perigee), but the method of setting the target orbit as apogee consumes the least fuel-a larger initial velocity represents the need for higher initial kinetic energy. and going to the determined position at the right initial velocity can naturally save energy.

It is the same to extend the above double-tangent orbit to different planets, which takes a position in the orbit of the inner planet as the perihelion and a position in the orbit of the outer planet as the aphelion point. and the orbit which is tangent to the orbits of the two stars is called Horman transfer orbit, which is the most energy-saving route in interplanetary travel. Accelerate in the direction of the earth's revolution when flying to the outer planet, accelerate again when approaching the target planet, slow down in the opposite direction when going to the inner planet, and slow down again when approaching the target planet. Because many planets in the solar system are in constant motion and their relative positions with the earth are changing all the time, the best time for interplanetary vehicle launch is periodic, which is equivalent to the rendezvous period between the earth and the target planet.

Because the Horman transfer from Earth to Mars takes too long because of its long distance and slow speed, sometimes it seems that the loss outweighs the gain by wasting a lot of time just to save fuel, so we can also use a parabolic orbit, that is, we need to reach the escape velocity of the sun in Earth's orbit (relative to the initial velocity of Earth is 16.7 km / s, of course) and spend more fuel to accelerate and decelerate during the departure and arrival phases. The following table shows the time it takes to get to the planets, and it can be seen that the parabolic orbit does take very short time compared with the double-cut Horman transfer orbit.

The comparison of the time required between Horman orbit and parabolic orbit is finally worth mentioning that the orbits of the major planets are not exactly round and their inclination angles are different. not only that, during the flight of the interplanetary vehicle, it is necessary to calculate the gravitational perturbation of the planets outside the target, even the effects of solar outbursts and the safety of passing through the asteroid belt must be considered. Of course, the real track design work can not be described clearly in a few words, this article will only do these brief introduction, the right to throw a brick to attract jade.

This article comes from the official account of Wechat: APC Science Alliance (ID:apcscience), author: APC Jun

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