§ 3.2 Theoretical Transportation Energy
 § 3.2.1 Moving around in the Earth-Moon system
The Earth is 81 TIMES as massive as the Moon, and asteroids have a trivial mass compared to the Moon.
To get from the surface of the Earth or the Moon into orbit around Earth (where space products providing valuable space services will reside) requires energy in two forms:
- Energy to rise above the surface, i.e., "potential energy"; and
- Energy to stay up there without falling back down, that is, acquiring the speed to go into a circular orbit, i.e., "kinetic energy". In a circular orbit, gravity is countered by the centrifugal force constantly.
Notably, when the Space Shuttle goes into low Earth orbit about 500 kilometers up, only about 7% of the theoretical energy required goes into lifting it to that height (potential energy). About 93% of the energy goes into accelerating the Space Shuttle to a speed where it goes into a circular orbit (kinetic energy).
The total amount of energy (kinetic plus potential) required is often expressed in terms of an analogy -- an "energy well", as pictured here, as if each gravitational body represented a hole in the ground like a water well which a cargo must crawl out of. The bigger the planet, the deeper the equivalent well. In the picture below, the vertical "height" represents the energy required to move from one point to the other, whereby the horizontal length represents the physical distances (to scale).
Note on the graph that the energy required to go into "geostationary earth orbit (GEO)", i.e., "stationary communications satellite orbit", from the Moon is small, compared to coming from Earth. It will later be shown that the energy required to get material from many asteroids near Earth into geosynchronous orbit is even less than from the Moon.
The Space Shuttle can goes to about 500 kilometers, and doesn't have the capability to go significantly higher than that, energy-wise. Communications satellite orbit is at 36,000 kilometers. Roughly half the energy to get to geosynchronous orbit is consumed in just getting to an orbital speed.
When the Space Shuttle carries a communications satellite up, it brings it only above the atmosphere to the 500 kilometer orbit. From there, the satellite is removed from the cargo bay and then launches to geosynchronous orbit 36,000 kilometers up using its own fuel propellant, which mades up most of the cargo in the Shuttle bay, not the satellite. But this is another issue for another place.
What orbital speeds are we talking about? For low Earth orbit, we are looking at a little over 7 kilometers per second (i.e., about 15,000 miles per hour), for the orbital speed. At this speed, the Shuttle orbits the Earth in about one and a half hours.
As a satellite goes higher in orbit where Earth's gravity is weaker, it does not need to go as fast to stay in orbit, and thus one orbit of the Earth takes much longer, e.g., 24 hours for GEO. However, it takes much more energy to lift it up to that orbit.
An orbit used by communications satellites is a high Earth orbit called "geostationary" or "geosynchronous" orbit, where it takes exactly 24 hours for one orbit. Since the Earth rotates once per 24 hours, each satellite stays "stationary" or "synchronized" above one point on Earth. That's why you can point your satellite TV dish to one place and leave it there, rather than having to track the satellite and lose communication if it were to pass over the horizon.
It takes more than 10 times more energy, theoretically, to get into geosynchrous Earth orbit from the surface of the Earth than from the surface of the Moon (that is, a circular orbit). The energy required from asteriods near Earth could be less or more than from the Moon, depending on the particular asteroid's orbital properties. Adding in the heavy vehicle and complexity associated with Earth launch, and launching the fuel for later in the flight, getting materials off of the Moon and especially from asteroids is much easier than from Earth.
To escape Earth orbit altogether takes less than 10% more energy than getting to geostationary orbit. Hence, the energy difference between GEO and other bodies besides Earth is often much less.
(One item often quoted by others it that it takes about 22 times more energy to launch from Earth and "escape" to infinity (without going into orbit) than to likewise launch from the Moon and escape to infinity. That is a simple comparison for laymen to illustrate the point, and differs somewhat from the more detailed comparison given here which accounts for getting into various useful circular orbits.)
It's important to understand that it takes just as much energy to come down as it does to go up -- there's no "free downhill". Coming "downhill" takes just as much energy and fuel in space because there is no friction -- you must spend fuel to lower yourself into a circular orbit. (An exception could be "aerobraking", i.e., using the Earth's atmosphere for friction, but no such vehicle has been operated to date except for return to Earth's surface. Aerobraking is discussed in the vehicles section.) Without aerobraking, if you simply brake and fall down in an elliptical orbit, you'll soon be right back at the top of that elliptical orbit and ready for another cycle. To stay at the bottom of the orbit requires that you circularize your orbit when you arrive there by spending more fuel.
Higher orbits have more potential energy but less kinetic energy. In fact, mathematically, to move from a lower orbit to a higher orbit requires spending two parts potential energy for every one part kinetic energy reduced.
Notably, there's no energy shortcut -- if you skip going into an interim orbit but just shoot from the surface of the Earth to a high orbit, you don't save anything, theoretically. However, in practical terms, there are differences between trajectories to get into a circular high orbit so that you can spend significantly more than the theoretical minimum. In general, haste makes waste, in terms of energy and fuel spent. The theoretically best trajectory from Earth's surface to any Earth orbit is to first get into orbital space, so that one isn't fighting against gravity's pull back down, and then to spiral up slowly, thrusting perpendicular to the line of sight with Earth (i.e., adding purely centrifugal force). However, this is rarely followed due to economic factors other than fuel launched (e.g., time and complexity, and radiation belt damage factors).
On the graph: "Sea Level Earth Orbit (SLEO)" just illustrates the minimum energy required to "stay in outer space" rather than standing (or crashing back) on Earth's surface -- Sea Level Earth Orbit is, say, a purely theoretical orbit just one foot above sea level as if there were no atmosphere or hills to crash into. Energy-wise, Sea Level Orbit represents the 93% kinetic energy to get to Shuttle orbit from Earth's surface, as compared to the 7% to lift up above the atmosphere. The Moon also has a "sea level orbit", or since it has "Mares" instead of "Seas", it has a corresponding "Mare Level Orbit".
The entire graph represents the theoretical minimum amount of energy required. However, the more energy required, the more fuel must be lifted for use later. Thus, the rocket size and complexity increase well out of proportion to the theoretical minimum energy required.
Asteroids have no significant escape velocity or "sea level orbital energy", and can be seen as objects already in orbital space. On the chart, they would be located beyond the dashed line above high Earth orbit, energy-wise. Some near Earth asteroids are just a tiny bit above the the dashed line, though most asteroids are significantly above the line. However, an analysis of retrieving asteroidal materials does not lend itself well to the above analysis, largely due to a concept called a "lunar gravity assist", which saves energy by trading orbital energy with the Moon, as discussed below.
§ 3.2.2 Asteroid materials retrieval
The delta-v's for known Earth-crossing asteroids are as low as 60 meters per second (60 m/s), as compared to the Moon's escape velocity of 2,400 m/s. There are many asteroids with required delta-v's lower than the lunar surface.
In a probable mission scenario to an asteroid, a large cargo will be launched into high Earth orbit and undergo a gravity assist by the Moon (discussed below) to pick up speed to rendezvous with the asteroid. After rendezvous of the cargo ship with the asteroid, any human presence needed would be sent by a small vehicle on a quick trajectory.
Before we go into specific missions, we should cover a key topic for retrieving asteroidal materials: lunar gravity assists.
§ 3.2.2.1 Lunar gravity assists for asteroids
Some asteroid enthusiasts humorously see the Moon mainly as an object to offer gravity assists, not to mine the Moon.
A "gravity assist" entails using a fly-by with the Moon to divert the trajectory of a payload and to impart delta-v, saving large amounts of fuel. Almost all NASA probes to other planets have depended on gravity assists, e.g., passing by the Moon and the Earth one or more times on their way out, and sometimes other planets as well for the purpose of gravity assists. For example, Voyager more than doubled its speed when it passed Jupiter.
One or more lunar gravity assists, sometimes in concert with an Earth gravity assist, will be used to:
- deflect incoming asteroid cargos into a high Earth orbit (or towards another gravity assist), and
- to brake the asteroid.
A single lunar gravity assist is illustrated conceptually below.
The maximum braking the Moon can provide is about 2.2 km/sec, using a "double lunar gravity assist", whereby the asteroid passes by the Moon coming in, then past the Earth, then past the Moon again going back out. This would divert the asteroid by almost 90 degrees from its original path, and capture it into a highly elliptical Earth orbit. Subsequent gravity assists would insert it into a more circular orbit around Earth after which it would perform final small thrusting maneuvers to achieve its desired destination orbit.
Many asteroids require a delta-v of much less than 2.2 km/sec, and require only a single lunar gravity assist (not an Earth gravity assist) to be captured, and optionally additional lunar gravity assists to divert the asteroid into a more circular orbit.
Gravity assists improve the economics of retrieving asteroid payloads, as well as outbound missions, and greatly broadens the number of attractive asteroids.
(In this game of "orbital billiards", we are tapping a gravitational energy source as asteroid payloads exchange orbital momentum with the Moon and the Earth -- the asteroid slows down while the Moon speeds up. Because asteroids are so small compared to the Earth and Moon, the effects on the Moon and Earth are so small as to be immeasurable. It would take millions of captured asteroids to cause any detectable changes in the Moon's or Earth's orbits. It's like measuring the effects of mosquitoes hitting the Empire State Building -- significant to the mosquito, but not to the building.)
We probably would not want to bring a complete asteroid in, but instead a series of small cargo containers which are more easily maneuvered and pose no significant threat to Earth. Trajectories are something we know very precisely, well in advance, and there's no need to get too close to Earth. The abovementioned 2.265 km/sec gravity assist maneuver was based on approaching no closer than several thousand miles (kilometers) of Earth's surface in order to allay such concerns. (Some people have proposed using the Earth's atmosphere for "aerobraking", but that's not at all what we are talking about here. We won't ever need to alert any emergency rendezvous team for pure gravity assist maneuvers.) One would expect that a quick response rendezvous team would be set up to protect Earth in the long run against both man-made objects and naturally occurring asteroids and big rocks that pass by Earth. Already, military and civilian telescopes have detected big rocks and sizeable asteroids passing very close to Earth, including skimming the upper atmosphere. If any of these naturally occurring objects had hit Earth, it would cause a natural disaster, possibly to the entire planet, not a man-made disaster. Man-made capabilities can prevent natural planetary damage.
§ 3.2.2.2 Specific asteroid missions
In the late 1970s, many people thought that the ideas of asteroidal materials utilization had so much merit that equipment would be developed and missions would be embarked upon by NASA. This was naive, but it was good that they proceeded with these projections, as they are exemplary. However, some of the dates of the following missions are already past.
The Amor asteroid "Anteros" (1973EC) was projected to have equipment launched to it in late 1992. Rendezvous would happen in 1993 and the equipment would be running at full steam by early 1994. After a delta-v of 1.6 km/sec, the cargo was to be enroute to the Earth-Moon system. It was to arrive in 1995 where two lunar gravity assists and a fuel thrust "capture maneuver" of 0.3 km/sec at orbit perigee would have put it into a circular orbit between the Moon and the Earth. (The 0.3 km/sec could be lowered by a third lunar encounter if so desired, but 0.3 km/sec is so small that it may be worth a little haste.)
The Amor asteroid "Eros" offered essentially the same story. The launch date was scheduled for a year later, in 1993. The delta-v would have been 1.7 km/sec and would've taken two lunar gravity assists and a capture maneuver of 0.3 km/sec.
The investigators thought that further analysis of mission opportunities and trajectories could reduce the delta-v to near 1 km/sec for the above two asteroids. On their shoestring budget, they did a limited number of calculations, and getting a trajectory under 2 km/sec initial delta-v was deemed enough to move onto other issues like analysis of the equipment needed.
In the late 1970s, a few of the newly discovered asteroids were also analyzed for rendezvous, e.g., the Apollo asteroids 1976UA (delta-v of 0.61 km/sec), 1973EC (delta-v of 1.43 km/sec) and 1977HB (delta-v of 1.06 km/sec). These calcuations were made using 1970s computers and some remarkably persistent professionals.
Since this 1970s study, using more sensitive telescopes, many more attractive targets have been found, including the asteroid 1982DB, which needs a delta-v of a mere 0.06 km/sec (i.e., 60 meters per second, or 130 miles per hour) to be captured by the Earth-Moon system.
§ 3.2.3 The L1, L2, L3, L4 and L5 "Libration" points
There are two points called L1 and L2 on the chart on energy requirements. Their relevance is as follows:
Both L1 and L2 orbit the Earth in the same amount of time that the Moon orbits the Earth. L1 and L2 serve as lunar-stationary or lunar-synchronous points in space, just like communications satellites in geostationary orbit stay above one point on Earth. This is because the Moon keeps one face pointed towards the Earth all the time (i.e., its rotation period equals its orbit period exactly).
In other words, if you're on the surface of the Moon and you look up at an object stationed at the L1 or L2 points, it won't move over time. Objects everywhere else in orbital space will move.
The Mass Driver, discussed in the lunar launch section, would shoot payloads to a Mass Catcher located at one of these two points. Lunar material can be shot off the lunar surface and into circular orbit around Earth without being placed in lunar orbit first, and rockets could head there, too.
Likewise, a solar power station or giant mirror system at these two points could service the lunar surface during most of the long lunar night by beaming down energy, light and warmth at night, in a longer term scenario.
In reality, stationkeeping would be required to keep an object at L1 or L2 due to the effects of the Sun's gravity. The fuel propellant required for stationkeeping would be small.
The rest of this article covers the physics, not the relevancy, of the L1 and L2 phenomenon, as well as the L3, L4 and L5 points, for those who are curious to know. Notably, L4 and L5 are two candidate regions for emplacing industrial facilties in high Earth orbit. Indeed, a famous space advocacy organization promoting asteroidal and lunar materials utilization called itself the L-5 Society.
These points are also called the LaGrange points (after their mathematical discoverer) or the "libration points".
The physics of L1, L2, L3, L4 and L5
L1 and L2 are two unique points caused by the interaction of the Earth's and Moon's gravities. They are also shown in the following chart, along with L3, L4 and L5.
The L1 point is where:
the Earth's pull = the Moon's pull + the orbital centrifugal "force"
When an object placed between them is balanced by two pulls in the two directions, it's as if the object is balanced at the peak of a hill. A nudge one way or the other would cause the object to fall inward towards Earth or outward towards the Moon. (It's actually a little more involved than this in that nothing falls straight down in a three-body rotating system, but you get the idea.)
The L2 point is another place where you can place an object and it will always stay there if balanced perfectly. The cause of the L2 point is: the Earth's gravity + the Moon's gravity = the orbital centrifugal force
The L3 point is simply the place in Earth orbit opposite the Moon which orbits the Earth in the same period of the Moon. It has no relevant economic potential (unless you never want to see the Moon for some reason).
The L4 and L5 "points" actually denote the centers of two regions -- objects will tend to drift around these two points in erratic orbits (appearing to orbit the empty points in a noncircular and nonelliptical shape) , without leaving the L4 or L5 region, and without requiring much stationkeeping propellant to stay in the region. The L4 and L5 points are located in Earth orbit 60 degrees in front of and 60 degrees behind the path of the Moon around the Earth. What happens is that when the Moon accelerates/decelerates objects in L4/L5 region, it changes their centrifugal force relative to Earth and hence their orbit around Earth, causing them to climb away from or fall towards Earth, in which case their orbital speed decreases or increases as they climb away or fall towards Earth, and as a result they fall behind or pass up the L4/L5 point and the cycle repeats. (Another way of looking at this is that an object placed at L4 or L5 orbits the center of mass of the Earth-Moon system in one month.)
Like everywhere else, the Sun's gravity perturbs objects in the L4 and L5 points so that it's not a perfect picture, but it's fairly close.
In the 1970s, many people saw the L4 and L5 regions as favorable for placing industry to process asteroidal and lunar material, and space colonies as well, because they all stayed in proximity to each other, unlike objects in different orbits outside this zone (unless the latter are all in the same orbit and just strung out in a circular curve), and stayed "close" to the Moon in terms of energy.
Whether or not space industry is located at L-4 or L-5 is a minor issue. Space industry could be located in most any Earth orbit. But the L-4 and L-5 points are referenced by many studies as candidate sites, hence their discussion here.
Different studies put the manufacturing facilities in different places, e.g., in a high orbit or an L point. Few put them in a low Earth orbit, due to the energy required to bring asteroidal materials down. Only some finished products come down. Wherever the industry is located, the initial large colonies will be located nearby.
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