How Do We Send Spacecraft to Other Planets? A Brief Guide to Motion and Orbits

Author: 丛雨
Reviewer: 时光

How many steps does it take to send a probe to Mars? In general, the flight of a probe launched from Earth toward another planet can be divided into three stages: the rocket launches and enters Earth orbit; the spacecraft accelerates onto a new trajectory, escapes Earth’s gravity, and travels toward the target planet; and, after arrival, it either enters orbit around that planet or lands on it. The first and third stages are short but involve many intricate steps and details, while the second takes up most of the journey. Starting from the problems of spacecraft motion and orbital trajectories, this article introduces some basic ideas about celestial motion and orbital maneuvers.

The Three Cosmic Velocities

We will begin with spacecraft speed.

As everyone knows, the faster an object is thrown horizontally, the farther away it lands. When its speed reaches 7.9 km/s, it no longer falls back to the ground; under the influence of Earth’s gravity alone, it travels around Earth in uniform circular motion. This is Earth’s first cosmic velocity: the minimum initial speed needed to launch an artificial satellite and the maximum linear speed for a circular orbit around Earth.

If the initial launch speed reaches approximately 11.2 km/s—Earth’s second cosmic velocity—the spacecraft can just escape Earth’s gravity completely without any further acceleration. A celestial body’s second cosmic velocity is also called its escape velocity. An unpowered object launched in any direction from the body’s surface at escape velocity gradually slows under gravity, reaching a speed of exactly $0$ at infinity. Escape velocity is $\sqrt{2}$ times the first cosmic velocity. It can be calculated from the conversion between kinetic energy and gravitational potential energy ($E_p = -GMm/r$, taking the potential energy at infinity as zero). During the escape, kinetic energy is gradually converted into gravitational potential energy until both reach $0$ at infinity.

The third cosmic velocity is the minimum initial launch speed required to escape the gravity of both Earth and the Sun. Unlike the second cosmic velocity, which may point in any direction, the third requires the spacecraft initially to travel in the same direction as Earth’s revolution around the Sun. This makes use of Earth’s orbital velocity and reduces fuel consumption. As explained above, the Sun’s escape velocity at Earth’s orbit is $\sqrt{2}$ times the circular-orbit speed of 29.8 km/s, or about 42.1 km/s. A spacecraft therefore needs the kinetic energy corresponding to the 12.3 km/s difference between those values to escape the Sun, as well as the kinetic energy corresponding to another 11.2 km/s to escape Earth. The speed corresponding to the sum of these two kinetic energies is the third cosmic velocity, 16.7 km/s.

Interplanetary spacecraft traveling between Earth and other planets plainly have destinations far enough away that they must, in a sense, overcome Earth’s gravity completely, yet they do not leave the Sun’s gravitational domain. Their speeds therefore usually lie between $v_2$ and $v_3$.

Deriving the second and third cosmic velocities

Orbits at Different Speeds

Next, we consider the particular orbital shapes produced by different speeds and launch directions.

The simplest case is an object launched horizontally. At the first cosmic velocity $v_1$, its orbit is circular. If the speed is slightly greater, the object’s centripetal acceleration will exceed what gravity provides, and its orbit becomes an ellipse, with Earth’s center at one focus and the launch point at periapsis. As the initial speed increases, the ellipse grows increasingly elongated, while its apoapsis altitude, semimajor axis, and eccentricity all increase. If the object is launched at the second cosmic velocity $v_2$, the eccentricity becomes 1 and both the semimajor axis and apoapsis distance become infinite, producing an open parabolic trajectory. If the initial speed exceeds $v_2$, the spacecraft follows one branch of a hyperbola relative to Earth. Its eccentricity is greater than 1, and in celestial mechanics its semimajor axis is defined as negative.

Looking back at horizontal launches slower than $v_1$, their trajectories are, strictly speaking, also ellipses. When the initial speed is very low, however, the ellipse is likewise extremely elongated and its eccentricity approaches $1$, so the path can be treated as a parabola. In this case, the launch point is the ellipse’s apoapsis and its periapsis lies inside Earth.

Orbital shapes produced by different horizontal launch speeds

In fact, apart from the special case of a vertically upward launch, an object follows an elliptical, parabolic, or hyperbolic path when its initial speed is respectively less than, equal to, or greater than $v_2$, even if it is not launched horizontally. With different launch directions, however, the semimajor axis and eccentricity of an ellipse or hyperbola, or the focal length of a parabola, will also differ. In these nonhorizontal cases, if the speed and the length of the radius vector are known, we can determine the shape and semimajor axis of the orbit by calculating the object’s mechanical energy. A sum of kinetic and potential energy that is less than, equal to, or greater than $0$ corresponds respectively to an elliptical, parabolic, or hyperbolic orbit. Because the mechanical energy of a two-body system is $E = -GMm/2a$, its semimajor axis $a$ can then be found.

Orbits produced by launches at v1 and v2 in different directions

Orbital Maneuvers and Hohmann Transfers

Finally, let us turn to orbital maneuvers. Real trajectory changes and mission designs are necessarily extremely complex, so the following discussion covers only a few simple principles and ideas.

Consider a spacecraft or satellite moving around Earth in a circular low Earth orbit. It suddenly fires its engine to accelerate in its direction of travel. Its speed is now greater than the circular-orbit speed at that point, and its centripetal acceleration exceeds what circular motion requires. The orbit becomes an ellipse whose periapsis is the point of acceleration, just like the surface-launch case discussed above in which the speed lies between $v_1$ and $v_2$. When the spacecraft reaches apoapsis, it accelerates by the appropriate amount once more in its direction of travel, bringing its speed exactly to the circular-orbit speed at that distance. It has now transferred successfully into a circular orbit with a larger radius and a greater altitude than the original one. This is the basic idea behind an orbital maneuver. Lowering an orbit requires only the reverse operation: decelerate into an elliptical orbit, then decelerate again at periapsis.

Diagram of a satellite changing orbit

Of course, the new orbit does not have to be circular. A single brief engine burn may not release enough energy to reach the target altitude. The satellite can wait until it completes an orbit and returns to the starting point before firing again, or it can perform a series of maneuvers, accelerating every time it reaches apoapsis. Its successive trajectories are then nested ellipses of different sizes. Acceleration and deceleration are not limited to apoapsis and periapsis, respectively, but making the target orbit the apoapsis uses the least fuel. A greater initial speed requires more initial kinetic energy, so using exactly the speed needed to reach a specified point naturally saves energy.

The same tangential-transfer principle can be extended to travel between planets. A trajectory that takes a point on an inner planet’s orbit as its perihelion and a point on an outer planet’s orbit as its aphelion, while remaining tangent to both planetary orbits, is called a Hohmann transfer orbit. It is the most energy-efficient route for interplanetary travel. To fly to an outer planet, a spacecraft accelerates in the direction of Earth’s orbital motion and accelerates again near the target planet to catch up with it. To travel to an inner planet, it decelerates opposite the direction of revolution and decelerates again near the target planet to remain there. Because the planets of the Solar System are constantly moving and their positions relative to Earth continually change, the best launch opportunities for an interplanetary spacecraft recur periodically. That period is the synodic period of Earth and the target planet.

A Hohmann transfer orbit from Earth to Mars

Because a Hohmann transfer covers a long route at low speed, it takes a great deal of time. In some situations, saving fuel at the cost of so much time may not be worthwhile. A parabolic trajectory can be used instead. This requires reaching the Sun’s escape velocity at Earth’s orbit—the initial velocity relative to Earth is, of course, 16.7 km/s—and spending more fuel on acceleration and deceleration during departure and arrival. The table below lists the travel times to the planets. Compared with a bitangential Hohmann transfer orbit, a parabolic trajectory is indeed much faster.

Travel times for Hohmann and parabolic trajectories

Finally, it is worth noting that the planets do not move in perfect circles and that their orbital inclinations differ. An interplanetary spacecraft’s route must also account for gravitational perturbations from planets other than its target, the effects of solar eruptions, and even the safety of crossing the asteroid belt. Real trajectory design clearly cannot be explained in a few sentences. This article offers only a brief introduction—a starting point for further study.