# Some Things That Affect Spacecraft Orbits

The derelict Chinese space station Tiangong-1 is expected to re-enter this weekend. The precise time and location aren’t easy to predict, and this has caused some concern among various members of the public. A space station is much bigger than a normal satellite, and it’s conceivable that larger pieces could cause major property damage or even injuries on impact. So why don’t we know where it will hit?

The answer is actually pretty complicated, because there many, many things which affect the paths of even simple, Earth-orbiting satellites. Some are immediately obvious, like drag from the upper atmosphere and Lunar perturbations. A lot are more subtle, like interference from Earth’s magnetic field. In this post I’d like to briefly explain all the significant factors, without going into too much math. Hopefully this will emphasize to specialists and laymen alike the value of responsible debris mitigation and the importance of safely disposing old spacecraft before ground controllers lose contact.

Atmospheric drag is, of course, the largest force affecting satellite orbits, and the reason Tiangong-1 is re-entering. As I explained in my last post about orbital mechanics, there is no clear upper edge to the atmosphere. Below about 500 kilometers, the atmosphere is still sufficiently thick to de-orbit most debris. Satellites above this altitude can remain in space for decades without any maintenance, and geosynchronous orbiters will last centuries or even millennia.

Low Earth Orbit is the natural regime to place crewed spacecraft, since the debris risk is dramatically lower. (The radiation hazards are also significantly higher in the medium orbits, and our current manned capsules can’t easily punch through the Van Allen belts.) This is fine as long as the spacecraft is operational and performing orbital maintenance maneuvers. The International Space Station has to perform these regularly.

ISS altitude over the last year. Discontinuities between decay regions represent maneuvers to raise the orbital height.

Source: heavens-above.com

Atmospheric drag on spacecraft is governed by a simple equation, which will be extremely familiar to even underclassmen aerospace engineers:

$F_D = \frac{1}{2} m C_D \rho A V^2$

In essence, this equation states that drag force $F_D$ on the spacecraft is equal to half the product of the spacecraft mass $m$, cross-sectional area $A$, atmospheric density $\rho$ and the square of the relative velocity $V^2$, all multiplied by a fudge factor $C_D$. We call the fudge factor the drag coefficient, and for satellites it tends to fall between 2.1 and 2.3, though higher values are common depending on the spacecraft configuration and materials used.

Density is the biggest unknown factor. It varies based on the altitude, time of day, time of the year, and how active the Sun has been lately. A number of these factors also affect relative velocity, because the atmosphere isn’t still. Upper atmospheric winds can be significant, often hundreds of meters per second.

Altitude plays a role, because different gases have different masses.  Mass is related to velocity via kinetic energy and the ideal gas law, which essentially states that the velocity of a particle, for a given internal energy, with be greater if the particle has a lower mass. Essentially, heavier gases like molecular oxygen and nitrogen will hug the planet, while lighter gases such as hydrogen and helium can easily climb into the thermosphere.

Additionally, the amount of solar and cosmic radiation is higher the further one rises. Collisions with energetic particles cause molecules to break into their components, so the frequency of atomic oxygen and nitrogen increases dramatically. This affects the effective density of the upper atmosphere and therefore the drag force. Since the amount of solar radiation is highly variable over the short term and changes with the eleven-year solar cycle, the numerical density of the upper atmosphere isn’t at all constant.

Solar radiation was ultimately responsible for the failure of a previous space station: Skylab. Initially NASA expected it to remain on-orbit till the early 1980s, when the space shuttle could have pushed it higher. The sun was particularly active during that era, however, and by the mid-1970s it was clear that the station would re-enter much earlier. A special mission would have saved it, but NASA never seriously planned one and Congress never allocated any funds. Skylab disintegrated over Australia on July 11, 1979. Despite widespread concerns, there was very little damage and no one was hurt, though one jurisdiction fined NASA \$400 for littering.

Skylab photographed by the last departing crew in February 1974.

Source: NASA

Temperature changes over the course of the day and year also affect the upper atmospheric density. Warm gases move faster and rise higher, and which can be seen in aggregate as an increase in pressure. Warm air near the surface displaces upwards, effectively increasing the size of the atmosphere. This can significantly increase the drag which a satellite experiences.

Generally speaking, day is warmer than night. Summer is warmer than winter, though of course that depends on which part of Earth you’re considering. (Remember that seasons are caused by axial tilt, not the eccentricity of our path around the Sun.) Advanced density models take all of these factors into consideration—date, time of day, latitude, longitude, altitude, and multiple indices of solar activity.

Atmospheric density is the biggest effect that astrodynamicists have to consider, especially for satellites in low orbits, but there are several other major forces which can cause significant accelerations, particularly upon smaller objects. From Newton’s Second Law, the acceleration a force has on an object is directly related to the mass of the object in question. Space stations have high masses, so only large forces can affect them. Cubesats have much lower masses, and can see large perturbations from forces that space station controllers happily ignore.

One of the larger effects we have to worry about is Earth oblateness, usually abbreviated as $J_2$. This is a low-order model of Earth’s shape, which is actually really complicated. High-fidelity models often reach high two-digit orders, though these are only needed for the most precise of missions.

The effect of $J_2$ is to reduce the inclination of an orbit, while also causing the orbit to change its orientation around Earth. The essential mechanism is that, because Earth is not exactly a sphere, when the spacecraft is north of the equator, it will be experienced an unbalanced force pulling it south, which the reverse occurs when it passes below the equator again. Over time, the satellite trajectory will approach an equatorial orbit, precessing around the pole.

We can define a $J_2$ value for every planet, because their non-sphericity is caused by rotation. Other gravitational effects may also be present, though these can usually be neglected. Luna is a notable exception. The internal mass distribution of our moon is so irregular than only a handful of orbital inclinations are stable. We learned this the hard way when Apollo astronauts left hand-launched microsatellites in orbit. One lasted for over a year, while other crashed within weeks.

This alludes to another important point: anything the size and density of a planet is not going to be perfectly rigid. In addition to ocean tides, gravitational effects from the Sun and Luna also cause internal tides, as the solid material within the planet bends slightly. These are called solid Earth tides, and affect satellite orbits about an order of magnitude more than the ocean tides. Usually these can be neglected, but precision gravity models need to taken them into account.

Accelerations of about the same order of magnitude ($\approx 10^{-12} \ \textrm{km}/\textrm{s}^2$) arise from general relativity. Because the spacecraft is travelling around a deep planetary gravity well, there will actually be a small but measurable change in velocity over time due to the spacecraft’s velocity through space. Mathematically, this is stated as:

$a_r = -3\frac{\mu}{r^2}\frac{v^2}{c^2}$

where $a_r$ is the relativistic acceleration, $\mu$ is the standard gravitational parameter for the body in question, $r$ is the distance of the spacecraft from the body, $v$ is the instantaneous velocity of the spacecraft, and $c$ is the speed of light. This acceleration will get smaller the further the spacecraft is from the primary body, and the slower it is moving (which, incidentally, spacecraft in higher orbits tend to do).

Generally, the position effects of relativity will be on the order of a centimeter, so orbit determination on larger scales can neglect that easily. Spacecraft in higher orbits probably won’t be affected, though spacecraft in very low orbits may see significant effects if they remain in space for a long time. The planet Mercury, orbiting so close to the massive Sun, experiences a dramatic relativistic acceleration. Astronomers once suspected that another planet lurked close to the Sun, perturbing Mercury’s orbit in a more typical way, until Einstein’s theory closely predicted the observed variations.

The Sun and Luna cause the greatest gravitational perturbations to Earth orbiting satellites, dragging the satellite slightly astray as it swings around our green planet. If we want to be really precise, though, we need to consider accelerations from other worlds, as well. Astrodynamicists rarely find a case where any besides Venus and Jupiter can be worth the trouble (and even those rarely). Venus is important because it is close to Earth, while faraway Jupiter is merely massive. Mars is nearly as close as Venus during much of its orbit, but is so much less massive that we neglect it entirely.

There’s one final type of force which affects orbits directly: radiation pressure. This can be somewhat counter-intuitive, because the majority of the radiation in question comes in the form of massless photons. $F = ma$ would seem to break down. Massless though they be, photons carry momentum. The rigorous definition of Newton’s Second Law of Motion is really that force is the change in momentum, which is the product of mass and velocity. We simply assume that mass remains constant in most applications.

Since photons carry momentum, however, they can produce a force directly. It’s not much, but when you consider how many photons are streaming through space each second, it turns out to be significant. The radiation flux from the Sun is usually around 1376 $\textrm{W}/\textrm{m}^2$, while the reflected radiation from Earth is around 459 $\textrm{W}/\textrm{m}^2$. Dividing by the speed of light, we find the two radiation pressures.

The force $F_s$ from this pressure $P_s$ is a function of the cross-sectional area of the spacecraft and the angle $\alpha$ between it and the incoming photons:

$F_s = P_s\cos\alpha$

The same equation can be used to calculate the effect of Earth radiation pressure, though the incidence angle will necessarily be different.

At this point, let’s return to the question of Tiangong-1. We’ve seen a great many reasons why it is challenging to make long-term predictions about its orbit. However, there is another problem which affects derelict spacecraft in particular: we don’t know its orientation in space. Now a large spacecraft can be imaged by radar, which gives us some idea about its orientation, but that will not work for smaller satellites. To make matters worse, it turns out Tiangong-1 is rotating, so its orientation will be different depending on when we look.

“What causes this rotation?” you might be wondering. I’m glad you asked. There are four major disturbance torques that attitude control engineers have to consider, though several of the above forces can also cause smaller torques.

At low altitudes, the largest torque by far is aerodynamic torque $T_a$. Recall the drag force equation I introduced above. Any offset between the center of the cross-sectional area and the center of mass will produce a moment arm $L$, over which the the force can act to induce rotation:

$T_a = F_D L$

We again find ourselves facing the fact that we may not know the cross-sectional area. If the precise forces and and orientation are well-known when ground controllers lose contact, we can model its rotation forward for a time, but not long enough to predict the re-entry of Tiangong-1.

But that’s not the end of it. There’s also force from Earth’s magnetic field, which will align any magnetized object with it, including spacecraft. The magnetic torque $T_m$ is a function of the residual magnetic dipole $M$ on the spacecraft, the Earth’s magnetic field strength $B$, and the misalignment $\theta$ between the two:

$T_m = MB\sin\theta$

The magnetic field strength is a function of altitude $r$ and latitude $l$, as follows:

$B = \frac{B_0 r_0^3}{r^3}\sqrt{3sin^2l + 1}$

where $B_0$ is the magnetic field strength at sea level ($3\times 10^{-5}\ \textrm{T}$) and $r_0$ is the average radius of Earth (about $6,370\ \textrm{km}$).

Finally, the spacecraft orientation leads to torques due to the small gravity gradient between its highest and lowest components. Measuring gravity gradient torque $T_g$ requires very accurate information about the moment of inertia distribution, which is often difficult to determine without detailed blueprints. It is also a function of altitude above the planet, because gravitation force (and therefore torque) drops off with increasing distance, and the angle $\theta$ off vertical”

$T_g = \frac{3\mu}{r^3}\lvert I_z - I_y\rvert\theta$

where $I_y$ and $I_z$ are the moments of inertia about the $y$– and $z$-axes, respectively.

As a general rule, the aerodynamic and magnetic torques dominate at lower altitudes, but may be significant for higher orbits or with special spacecraft configurations. These have to be determined on a case-by-case basis.

So that’s why we don’t know exactly when or where Tiangong-1 will re-enter. There’s a lot of variables involved, many of which are difficult to estimate, and often are strangely coupled at higher orders. There are many things which can be improved if we’re willing to spend enough money, such as building better atmospheric and gravitational models or tracking debris more regularly. Ultimately, though, there’s no substitute for responsible disposal at the end of a spacecraft’s mission, either into a stable graveyard orbit or by de-orbiting at a safe location.

…and all of this assumes that the spacecraft generates no thrust. Imagine how complex it gets when we consider outgassing!

### References

• Brown, C. D., Dukes, E. M., Elements of Spacecraft Design, edited by J. S. Przemieniecki, AIAA Education Series, AIAA, New York, 2003.
• Knipp, D. J., Understanding Space Weather and the Physics Behind It, 1st ed., McGraw Hill, New York, 2011.
• Moe, K., Moe, M. M., “Gas-surface interactions and satellite drag coefficients,” Planetary and Space Science, Vol. 53, No. 8, 2005, pp. 793–801
• Montenbruck, O., Gill, E., Satellite Orbits: Models, Methods, and Applications, 1st ed., Springer-Verlag, Berlin, 2000.
• Vallado, D. A., Fundamentals of Astrodynamics and Applications, 4th ed., Microcosm, Hawthorne CA, 2013.

# What Constitutes Space?

I’ve been writing about the assorted difficulties faced in astronautical engineering, but this presupposes a certain amount of background knowledge and was quickly getting out of hand. So let’s start with a simpler question: what is space, anyway?

Generally speaking, space is the zone beyond Earth’s atmosphere. This definition is problematic, however, because there’s no clean boundary between air and space. The US Standard Atmosphere goes up to 1000 km. The exosphere extends as high as 10,000 km. Yet many satellites (including the International Space Station) orbit much lower, and the conventional altitude considered to set the edge of space is only 100 km, or 62.1 miles.

This figure comes from the Hungarian engineer Theodore von Kármán. Among his considerable aerodynamic work, he performed a rough calculation of the altitude at which an airplane would need to travel orbital velocity to generate sufficient lift to counteract gravity, i.e. the transition from aeronautics to astronautics. It will vary moderately due to atmospheric conditions and usually lies slightly above 100 km, but that number has been widely accepted as a useful definition for the edge of space.

To better understand this value, we need to understand just what an orbit is.

Objects don’t stay in space because they’re high up. (It’s relatively easy to reach space, but considerably harder to stay there.) The gravity of any planet, Earth included, varies with an inverse square law, that is, the force which Earth exerts on an object is proportional to the reciprocal of the distance squared. This principle is known as Newton’s Law of Universal Gravitation. Its significance for the astronautical engineer is that moving a few hundred kilometers off the surface of Earth results in only a modest reduction of downward acceleration due to gravity.

To stay at altitude, a spacecraft does not counteract gravity, as an aircraft does. Instead, it travels laterally at sufficient speed that the arc of its curve is equal to the curvature of Earth itself. An orbit is a path to fall around an entire planet.

The classic example to illustrate this concept, which also comes from Newton, is a tremendous cannon placed atop a tall mountain (Everest’s height was not computed until the 1850s). As you can verify at home, an object thrown faster will land further away from the launch point, despite the downward acceleration being identical. In the case of our cannon, a projectile shot faster will land further from the foot of the mountain. Fire the projectile faster enough, and it will travel around a significant fraction of the Earth’s curvature. Firing it fast enough1 and after awhile it will swing back around to shatter the cannon from behind.

Source: European Space Agency

In this light, von Kármán’s definition is genius. While there is no theoretical lower bound on orbital altitude2, below about 100 km travelling at orbital velocity will result in a net upwards acceleration due to aerodynamic lift. Vehicles travelling below this altitude will essentially behave as airplanes, balancing the forces of thrust, lift, weight and drag—whereas vehicles above it will travel like satellites, relying entirely on their momentum to stay aloft indefinitely.

But we should really give consideration to aerodynamic drag in our analysis, because it poses a more practical limit on the altitude at which spacecraft can operate. Drag is the reason you won’t find airplanes flying at orbital speeds in the mesosphere, and the reason satellites don’t orbit just above the Kármán line. Even in the upper atmosphere, drag reduce a spacecraft’s forward velocity and therefore its kinetic energy, forcing it to orbit at a lower altitude.

This applies to all satellites, but above a few hundred kilometers is largely negligible. Spacecraft in low Earth orbit will generally decay after a number of years without repositioning; the International Space Station requires regular burns to maintain altitude. At a certain point, this drag will deorbit a satellite within a matter of days or even hours.

The precise altitude will depend on atmospheric conditions, orbital eccentricity, and the size, shape, and orientation of the satellite, but generally we state that stable orbits are not possible below 130 kilometers. This assumes a much higher apoapsis: a circular orbit below 150 km will decay just as quick. To stay aloft indefinitely, either frequent propulsion or a much higher orbit will be necessary3.

On the other hand, it is exceedingly difficult to fly a conventional airplane above the stratosphere, and even the rocket-powered X-15 had trouble breaking 50 miles, which is the US Air Force’s chosen definition. Only two X-15 flights crossed the Kármán line.

Ultimately, then, what constitutes the edge of space? From a strict scientific standpoint, there is no explicit boundary, but there are many practical ones. Which one to chose will depend on what purposes your definition needs to address. However, von Kármán’s suggestion of 100 km has been widely accepted by most major organizations, including the Fédération Aéronautique Internationale and NASA. Aircraft will rarely climb this high and spacecraft will rarely orbit so low, but perhaps having few flights through the ambiguous zone helps keep things less confusing.

1For most manned spaceflights, this works out to about 7,700 meters per second. The precise value will depend on altitude: higher spacecraft orbit slower, and lower spacecraft must orbit faster4. In our cannon example, it would be a fair bit higher, neglecting air resistance.

2The practical lower bound, of course, is the planet’s surface. The Newtonian view of orbits, however, works on the assumption that each planet can be approximated as a single point. This isn’t precisely true—a planet’s gravitation force will vary with the internal distribution of its mass, which astrodynamicists exploit to maintenance the orbits of satellites. That, however, goes beyond the scope of this introduction.

3The International Space Station orbits so low in part because most debris below 500 km reenters the atmosphere within a few years, reducing the risk of collision. This is no trivial concern—later shuttle missions to service the Hubble Space Telescope, which orbits at about 540 km, were orchestrated around the dangers posed by space junk.

4Paradoxically, we burn forward to raise an orbit, speeding up to eventually slow down. This makes perfect sense when we consider the reciprocal relationship between kinetic and potential energy, but that’s another post.

# German Researcher Discovers Most Efficient Path to Mars

A civil engineer in Essen, Germany has determined the transfer orbit which will get astronauts to Mars the quickest.

Walter Hohmann, a civil engineer, spent several years studying physics and astronomy before publishing his book The Attainability of the Celestial Bodies. It may become required reading for NASA mission planners.

Fuel requirements will be central to the architecture of interplanetary spaceflights, Dr. Hohmann expects. To account for this, he solved for the trajectory which requires the least amount of velocity change, or what scientists call “delta V”. Spacecraft produce this acceleration by firing rocket engines.

The most efficient orbit between two planets turned out to be an ellipse that lies tangent to the planets’ orbital paths.

Source: University of Arizona

Such an orbit requires the least amount of energy to achieve when starting from Earth, but has a serious drawback. Least-energy trajectories are also the slowest. For a crewed mission, taking along enough food and oxygen could make a less efficient path ultimately cheaper.

Another problem is waiting for planets to be in the right place for launch. Because Earth orbits the sun faster than the outer planets and slower than the inner planets, the possible alignment for such a transfer trajectory only occurs occasionally. The window to leave for Mars only opens every two years, for example. Launching interplanetary spacecraft at other times would require vastly more fuel.

Nevertheless, astronomers and aerospace engineers find Dr. Hohmann’s discovery extremely useful for designing space missions.

Happy Amazing Breakthrough Day!