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Orbital elevator. Research work "space elevator" What is a space elevator

Orbital elevator. Research work "space elevator" What is a space elevator

(GSO) due to centrifugal force. It rises along a cable, carrying a payload. When rising, the load will be accelerated due to the rotation of the Earth, which will allow it to be sent beyond the Earth’s gravity at a sufficiently high altitude.

The cable requires extremely high tensile strength combined with low density. According to theoretical calculations, carbon nanotubes seem to be a suitable material. If we assume their suitability for the manufacture of a cable, then the creation of a space elevator is a solvable engineering problem, although it requires the use of advanced developments and. The creation of the elevator is estimated at 7-12 billion US dollars. NASA is already funding related developments at the American Institute for Scientific Research, including the development of a lift capable of moving independently along a cable.

Design

There are several design options. Almost all of them include a base (base), cable (cable), lifts and counterweight.

Base

The base of a space elevator is the place on the surface of the planet where the cable is attached and the lifting of the cargo begins. It can be mobile, placed on an ocean-going vessel.

The advantage of a movable base is the ability to perform maneuvers to evade hurricanes and storms. The advantages of a stationary base are cheaper and more accessible energy sources, and the ability to reduce the length of the cable. The difference of a few kilometers of tether is relatively small, but can help reduce the required thickness of its middle part and the length of the part extending beyond geostationary orbit.

Cable

The cable must be made of a material with an extremely high tensile strength to specific gravity ratio. A space elevator will be economically justified if a cable with a density comparable to graphite and a strength of about 65-120 gigapascals can be produced on an industrial scale at a reasonable price.

For comparison, the strength of most types of steel is about 1 GPa, and even the strongest types are no more than 5 GPa, and steel is heavy. The much lighter Kevlar has a strength in the range of 2.6-4.1 GPa, and quartz fiber has a strength of up to 20 GPa and higher. The theoretical strength of diamond fibers may be a little [for how long?] higher.

The technology for weaving such fibers is still in its infancy.

According to some scientists, even carbon nanotubes will never be strong enough to make a space elevator cable.

Experiments by scientists from the University of Technology Sydney made it possible to create graphene paper. Sample tests are encouraging: the density of the material is five to six times lower than that of steel, while the tensile strength is ten times higher than that of carbon steel. At the same time, graphene is a good conductor of electric current, which allows it to be used to transmit power to a lift, as a contact bus.

Thickening the cable

The space elevator must support at least its own weight, which is considerable due to the length of the cable. Thickening on the one hand increases the strength of the cable, on the other, it adds its weight, and therefore the required strength. The load on it will vary in different places: in some cases, a section of the tether must support the weight of the segments below, in others it must withstand the centrifugal force that holds the upper parts of the tether in orbit. To satisfy this condition and to achieve optimality of the cable at each point, its thickness will be variable.

It can be shown that taking into account the Earth's gravity and centrifugal force (but not taking into account the smaller influence of the Moon and the Sun), the cross-section of the cable depending on the height will be described by the following formula:

Here is the cross-sectional area of the cable as a function of the distance from center Earth.

The formula uses the following constants:

This equation describes a tether whose thickness first increases exponentially, then its growth slows down at an altitude of several Earth radii, and then it becomes constant, eventually reaching geostationary orbit. After this, the thickness begins to decrease again.

Thus, the ratio of the cross-sectional areas of the cable at the base and at the GSO ( r= 42,164 km) is:

Substituting here the density and strength of steel and the diameter of the cable at the ground level of 1 cm, we get a diameter at the GSO level of several hundred kilometers, which means that steel and other materials familiar to us are unsuitable for building an elevator.

It follows that there are four ways to achieve a more reasonable cable thickness at the GSO level:

Another way is to make the base of the elevator movable. Moving even at a speed of 100 m/s will already give a gain in circular speed by 20% and reduce the cable length by 20-25%, which will make it lighter by 50 percent or more. If you “anchor” the cable on a supersonic plane or train, then the gain in cable mass will no longer be measured in percentages, but in dozens of times (but losses due to air resistance are not taken into account).

Counterweight

A counterweight can be created in two ways - by tying a heavy object (for example, an asteroid, a space settlement or a space dock) beyond geostationary orbit, or by extending the tether itself a significant distance beyond geostationary orbit. The second option has become more popular lately because it is easier to implement, and in addition, it is easier to launch loads to other planets from the end of an elongated cable, since it has a significant speed relative to the Earth.

Angular Momentum, Velocity and Tilt

The horizontal speed of each section of the cable increases with height in proportion to the distance to the center of the Earth, reaching the first escape velocity in geostationary orbit. Therefore, when lifting a load, he needs to gain additional angular momentum (horizontal speed).

Angular momentum is acquired due to the rotation of the Earth. At first, the lift moves slightly slower than the cable (Coriolis effect), thereby “slowing down” the cable and slightly deflecting it to the west. At an ascent speed of 200 km/h, the cable will tilt by 1 degree. The horizontal component of tension in a non-vertical cable pulls the load to the side, accelerating it in an easterly direction (see diagram) - due to this, the elevator acquires additional speed. According to Newton's third law, the cable slows down the Earth by a small amount.

At the same time, the influence of centrifugal force forces the cable to return to an energetically favorable vertical position, so that it will be in a state of stable equilibrium. If the elevator's center of gravity is always above the geostationary orbit, regardless of the speed of the elevators, it will not fall.

By the time the cargo reaches the GEO, its angular momentum (horizontal velocity) is sufficient to launch the cargo into orbit.

When lowering the load, the reverse process will occur, tilting the cable to the east.

Launch into space

At the end of the cable at an altitude of 144,000 km, the tangential component of the speed will be 10.93 km/s, which is more than enough to leave the Earth's gravitational field and launch ships to Saturn. If the object is allowed to slide freely along the top of the tether, it will have enough speed to escape the solar system. This will happen due to the transition of the total angular momentum of the cable (and the Earth) into the speed of the launched object.

To achieve even greater speeds, you can lengthen the cable or accelerate the load using electromagnetism.

Construction

Construction is carried out from a geostationary station. This is the only place where a spacecraft can land. One end descends to the surface of the Earth, stretched by the force of gravity. The other, for balancing, is in the opposite direction, being pulled by centrifugal force. This means that all materials for construction must be lifted into geostationary orbit in the traditional way, regardless of the cargo's destination. That is, the cost of lifting the entire space elevator into geostationary orbit is the minimum price of the project.

Savings from using a space elevator

Presumably, the space elevator will greatly reduce the cost of sending cargo into space. Space elevators are expensive to build, but their operating costs are low, so they are best used over long periods of time for very large volumes of cargo. Currently, the market for launching loads may not be large enough to justify building an elevator, but the dramatic reduction in price should lead to greater variety of loads. Other transport infrastructure - highways and railways - justifies itself in the same way.

There is still no answer to the question whether the space elevator will return the money invested in it or whether it would be better to invest it in the further development of rocket technology.

We should not forget about the limit on the number of relay satellites in geostationary orbit: currently, international agreements allow 360 satellites - one relay per angular degree, in order to avoid interference when broadcasting in the K u -frequency band. For C frequencies the number of satellites is limited to 180.

This circumstance explains the real commercial failure of the project, since the main financial costs of non-governmental organizations are focused on relay satellites occupying either geostationary orbit (television, communications) or lower orbits (global positioning systems, natural resource observation, etc.) .

However, the elevator can be a hybrid project and, in addition to the function of delivering cargo into orbit, remain a base for other research and commercial programs not related to transport.

Achievements

Since 2005, the annual Space Elevator Games competition has been held in the United States, organized by the Spaceward Foundation with the support of NASA. There are two categories in these competitions: “best cable” and “best robot (lift)”.

In the lift competition, the robot must overcome a set distance, climbing a vertical cable at a speed not lower than that established by the rules (in the 2007 competition, the standards were as follows: cable length - 100 m, minimum speed - 2 m/s). The best result in 2007 was covering a distance of 100 m with an average speed of 1.8 m/s.

The total prize fund for the Space Elevator Games competition in 2009 was $4 million.

In the rope strength competition, participants must provide a two-meter ring made of heavy-duty material weighing no more than 2 grams, which a special installation tests for rupture. To win the competition, the strength of the cable must be at least 50% greater in this indicator than the sample already available to NASA. So far, the best result belongs to the cable that withstood a load of up to 0.72 tons.

The competition does not include Liftport Group, which gained notoriety for its claims to launch a space elevator in 2018 (later pushed back to 2031). Liftport conducts its own experiments, for example, in 2006, a robotic lift climbed a strong rope stretched with the help of balloons. Out of one and a half kilometers, the lift managed to cover only 460 meters. In August-September 2012, the company launched a project to raise funds for new experiments with the lift on the Kickstarter website. Depending on the amount collected, it is planned to lift the robot 2 or more kilometers.

At the Space Elevator Games competition, from November 4 to November 6, 2009, a competition organized by the Spaceward Foundation and NASA took place in Southern California, at the Dryden Flight Research Center, within the boundaries of the famous Edwards Air Force Base. The test length of the cable was 900 meters, the cable was lifted using a helicopter. The leadership was taken by LaserMotive, which presented a lift with a speed of 3.95 m/s, which is very close to the required speed. The elevator covered the entire length of the cable in 3 minutes 49 seconds; the elevator carried a payload of 0.4 kg. .

Similar projects

The space elevator is not the only project that uses tethers to launch satellites into orbit. One such project is the Orbital Skyhook. Skyhook uses a tether that is not very long compared to a space elevator, which is in low Earth orbit and rotates quickly around its middle part. Due to this, one end of the cable moves relative to the Earth at a relatively low speed, and loads from hypersonic aircraft can be suspended from it. At the same time, the Skyhook design works like a giant flywheel - an accumulator of torque and kinetic energy. The advantage of the Skyhook project is its feasibility using existing technologies. The downside is that Skyhook uses energy from its motion to launch satellites, and this energy will need to be replenished somehow.

Space elevator in various works

In the 1972 USSR film Petka in Space, the main character invents a space elevator.
One of Arthur C. Clarke's famous works, The Fountains of Heaven, is based on the idea of a space elevator. In addition, the space elevator appears in the final part of his famous tetralogy, A Space Odyssey (3001: The Last Odyssey).
In Star Trek: Voyager episode 3x19 "Rise", a space elevator helps the crew escape a planet with a dangerous atmosphere.
Civilization IV has a space elevator. There he is one of the later “Great Miracles”.
Timothy Zahn's science fiction novel Spinneret (1985) mentions a planet capable of producing superfiber. One of the races, interested in the planet, wanted to get this fiber specifically for the construction of a space elevator.
In Frank Schätzing's science fiction novel Limit, a space elevator acts as a central point of political intrigue in the near future.
In Sergei Lukyanenko’s dilogy “Stars are Cold Toys,” one of the extraterrestrial civilizations, in the process of interstellar trade, delivered heavy-duty threads to Earth that could be used to build a space elevator. But extraterrestrial civilizations insisted exclusively on using them for their intended purpose - to help during childbirth.
In the science fiction novel “Destined to Victory” by J. Scalzi (eng. Scalzi, John. Old Man's War) space elevator systems are actively used on Earth, numerous terrestrial colonies and some planets of other highly developed intelligent races for communication with the piers of interstellar ships.
In the science fiction novel “Tomorrow Will Be Eternity” by Alexander Gromov, the plot is built around the fact of the existence of a space elevator. There are two devices - a source and a receiver, which, using an “energy beam”, are capable of raising the elevator “cabin” into orbit.
Alastair Reynolds' science fiction novel "Abyss City" gives a detailed description of the structure and functioning of the space elevator and describes the process of its destruction (as a result of a terrorist attack).
Terry Pratchett's science fiction novel Strata features the Line, an extremely long artificial molecule used as a space elevator.
Mentioned in the song of the group Zvuki Mu “Elevator to Heaven”.
At the very beginning of the Sonic Colors game, Sonic and Tails can be seen taking the space elevator to get to Dr. Eggman's Park.
In Alexander Zorich’s book “Somnambulist 2” from the Ethnogenesis series, the main character Matvey Gumilyov (after planting a surrogate personality - Maskim Verkhovtsev, the personal pilot of comrade Alpha, the head of “Star Fighters”) travels in an orbital elevator.
In the story “Snake” by science fiction writer Alexander Gromov, the characters use a space elevator “on the way” from the Moon to the earth.
In George R. Martin's series of science fiction novels, "Tuf's Travels," on the planet "S"atlem, an orbital elevator leads to a planetoid equipped like a spaceport.

In manga and anime

In the third episode of the anime Edo Cyber City, a space elevator was used to ascend to the orbital cryogenic bank.
Battle Angel features a cyclopean space elevator, at one end of which is the Sky City of Salem (for citizens) along with a lower city (for non-citizens), and at the other end is the space city of Yeru. A similar structure is located on the other side of the Earth.
In the anime Mobile Suit Gundam 00, there are three space elevators; a ring of solar panels is also attached to them, which allows the space elevator to be used for generating electricity.
In the anime Z.O.E. Dolores features a space elevator, and also shows what could happen in the event of a terrorist attack.
The space elevator is mentioned in the anime series Trinity Blood, in which the Arc spaceship serves as a counterweight.

Notes

Literature

Yuri Artsutanov “Into space - on an electric locomotive”, newspaper “Komsomolskaya Pravda” dated July 31, 1960.
Alexander Bolonkin “Non-Rocket Space Launch and Flight”, Elsevier, 2006, 488 pgs.

Many people know the biblical story about how people set out to become like God and decided to erect a tower as high as heaven. The Lord, angry, made all the people speak different languages, and the construction stopped.

It’s hard to say whether this is true or not, but after thousands of years, humanity again thought about the possibility of building a supertower. After all, if you manage to build a structure tens of thousands of kilometers high, you can reduce the cost of delivering cargo into space by almost a thousand times! Space will once and for all cease to be something distant and unattainable.

Dear space

The concept of a space elevator was first considered by the great Russian scientist Konstantin Tsiolkovsky. He assumed that if you build a tower 40,000 kilometers high, the centrifugal force of our planet will hold the entire structure, preventing it from falling.

At first glance, this idea smells a mile away of Manilovism, but let's think logically. Today, most of the weight of rockets is fuel, which is spent on overcoming Earth's gravity. Of course, this also affects the launch price. The cost of delivering one kilogram of payload to low-Earth orbit is about $20,000.

So when relatives give jam to the astronauts on the ISS, you can be sure: this is the most expensive delicacy in the world. Even the Queen of England cannot afford this!

Launching one shuttle cost NASA between $500 and $700 million. Due to problems in the American economy, NASA management was forced to close the space shuttle program and outsource the function of delivering cargo to the ISS to private companies.

In addition to economic problems, there are also political ones. Due to disagreements over the Ukrainian issue, Western countries have introduced a number of sanctions and restrictions against Russia. Unfortunately, they also affected cooperation in astronautics. NASA received an order from the US government to freeze all joint projects, with the exception of the ISS. In response, Deputy Prime Minister Dmitry Rogozin said that Russia is not interested in participating in the ISS project after 2020 and intends to switch to other goals and objectives, such as establishing a permanent scientific base on the Moon and a manned flight to Mars.

Most likely, Russia will do this together with China, India and, possibly, Brazil. It should be noted: Russia was already going to complete work on the project, and Western sanctions simply accelerated this process.

Despite such grandiose plans, everything may remain on paper unless a more efficient and cheaper way to deliver cargo beyond the Earth's atmosphere is developed. A total of over 100 billion dollars were spent on the construction of the same ISS! It’s even scary to imagine how many “greenies” it will take to create a station on the Moon.

A space elevator could be the perfect solution to the problem. Once the elevator is operational, shipping costs could drop to two dollars per kilogram. But first you will have to thoroughly rack your brains about how to build it.

Margin of safety

In 1959, Leningrad engineer Yuri Nikolaevich Artsutanov developed the first working version of a space elevator. Since it is impossible to build an elevator from the bottom up due to the gravity of our planet, he proposed doing the opposite - building from the top down. To do this, a special satellite had to be launched into a geostationary orbit (about 36,000 kilometers), where it had to take a position above a certain point on the Earth's equator. Then begin assembling the cables on the satellite and gradually lower them towards the surface of the planet. The satellite itself also played the role of a counterweight, constantly keeping the cables taut.

The general public was able to become familiar with this idea in detail when, in 1960, Komsomolskaya Pravda published an interview with Artsutanov. The interview was also published by Western media, after which the whole world was subjected to “elevator fever.” Science fiction writers were especially zealous, painting rosy pictures of the future, an indispensable attribute of which was the space elevator.

All experts studying the possibility of creating an elevator agree that the main obstacle to the implementation of this plan is the lack of sufficiently strong material for the cables. According to calculations, this hypothetical material should withstand a voltage of 120 gigapascals, i.e. over 100,000 kilograms per square meter!

The strength of steel is approximately 2 gigapascals, for particularly strong options it is a maximum of 5 gigapascals, for quartz fiber it is slightly above 20. This is simply monstrously low. The eternal question arises: what to do? Develop nanotechnology. The most promising candidate for the role of an elevator cable may be carbon nanotubes. According to calculations, their strength should be much higher than the minimum 120 gigapascals.

So far, the strongest sample has been able to withstand a stress of 52 gigapascals, but in most other cases they have ruptured in the range of 30 to 50 gigapascals. In the course of lengthy research and experiments, specialists from the University of Southern California managed to achieve an unheard-of result: their tube was able to withstand a voltage of 98.9 gigapascals!

Unfortunately, this was a one-off success, and there is another significant problem with carbon nanotubes. Nicolas Pugno, a scientist from the Polytechnic University of Turin, came to a disappointing conclusion. It turns out that even due to the displacement of one atom in the structure of carbon tubes, the strength of a certain area can sharply decrease by 30%. And all this despite the fact that the longest nanotube sample obtained so far is only two centimeters. And if you take into account the fact that the length of the cable should be almost 40,000 kilometers, the task seems simply impossible.

Debris and storms

Another very serious problem is related to space debris. When humanity settled in low-Earth orbit, it began one of its most favorite pastimes - littering the surrounding space with the products of its vital activity. At the very beginning, we were somehow not particularly worried about this. “After all, space is endless! - we reasoned. “You throw away the piece of paper, and it will go on to explore the vastness of the Universe!”

This is where we made a mistake. All the debris and remains of aircraft are doomed to circle the Earth forever, captured by its powerful gravitational field. It doesn't take an engineer to figure out what would happen if one of these pieces of junk collided with a cable. Therefore, thousands of researchers from all over the world are racking their brains over the issue of eliminating a near-Earth landfill.

The situation with the base of the elevator on the surface of the planet is also not entirely clear. Initially, it was planned to create a stationary base at the equator to ensure synchronization with a geostationary satellite. However, then the harmful effects on the elevator of hurricane winds and other natural disasters cannot be avoided.

Then the idea came up to attach the base to a floating platform that could maneuver and “avoid” storms. But in this case, operators in orbit and on the platform will be forced to perform all movements with surgical precision and absolute synchronization, otherwise the entire structure will go to hell.

Keep your chin up!

Despite all the difficulties and obstacles that lie on our thorny path to the stars, we should not hang our noses and throw this, without a doubt, unique project into the back burner. A space elevator is not a luxury, but a vital thing.

Without it, colonization of near space will become an extremely labor-intensive, expensive endeavor and can take many years. There are, of course, proposals to develop anti-gravity technologies, but this is too distant a prospect, and the elevator is needed in the next 20-30 years.

An elevator is necessary not only for lifting and lowering loads, but also as a “mega-sling.” With its help, it is possible to launch spaceships into interplanetary space without spending huge volumes of such precious fuel, which otherwise could be used to accelerate the ship. Of particular interest is the idea of using an elevator to clean the Earth of hazardous waste.

Let’s say that spent nuclear fuel from a nuclear power plant can be placed in sealed capsules, and then sent at direct fire towards the Sun, for which burning such a booger is a piece of cake.

But, oddly enough, the implementation of such an idea is, rather, not a question of economics or science, but of politics. We need to face the truth - not a single country in the world can independently cope with such a grandiose project. There is no way to do without international cooperation.

First of all, the participation of the United States, the European Union, China, Japan, India, Brazil and, of course, Russia is important. So, no matter how you look at it, you will have to sit down at the negotiating table and smoke the pipe of peace. Therefore, guys, let's live together, and everything will work out for us!

Adilet URAIMOV

Although the construction of a space elevator is already within our engineering capabilities, the passions around this structure have unfortunately subsided recently. The reason is that scientists have not yet been able to obtain the technology to produce carbon nanotubes of the required strength on an industrial scale.

The idea of launching cargo into orbit without rockets was proposed by the same person who founded theoretical cosmonautics - Konstantin Eduardovich Tsiolkovsky. Inspired by the Eiffel Tower he saw in Paris, he described his vision of a space elevator in the form of a tower of enormous height. Its top would just be in a geocentric orbit.

The elevator tower is based on strong materials that prevent compression - but modern ideas for space elevators still consider a version with cables that must be tensile strength. This idea was first proposed in 1959 by another Russian scientist, Yuri Nikolaevich Artsutanov. The first scientific work with detailed calculations on a space elevator in the form of a cable was published in 1975, and in 1979 Arthur C. Clarke popularized it in his work “The Fountains of Paradise.”

Although nanotubes are currently recognized as the strongest material, and the only one suitable for building an elevator in the form of a cable stretching from a geostationary satellite, the strength of nanotubes obtained in the laboratory is not yet sufficient to reach the calculated strength.

Theoretically, the strength of nanotubes should be more than 120 GPa, but in practice the highest elongation of a single-walled nanotube was 52 GPa, and on average they broke in the range of 30-50 GPa. A space elevator requires materials with a strength of 65-120 GPa.

Late last year, the largest American documentary film festival, DocNYC, screened the film Sky Line, which describes the attempts of US engineers to build a space elevator - including participants in the NASA X-Prize competition.

The main characters of the film are Bradley Edwards and Michael Lane. Edwards is an astrophysicist who has been working on the space elevator idea since 1998. Lane is an entrepreneur and founder of LiftPort, a company promoting the commercial use of carbon nanotubes.

In the late 90s and early 2000s, Edwards, having received grants from NASA, intensively developed the idea of a space elevator, calculating and evaluating all aspects of the project. All his calculations show that this idea is feasible - if only a fiber strong enough for the cable appears.

Edwards briefly partnered with LiftPort to seek funding for the elevator project, but due to internal disagreements, the project never materialized. LiftPort closed in 2007, although a year earlier it had successfully demonstrated a robot climbing a mile-long vertical cable suspended from balloons as part of a proof of concept for some of its technology.

That private space, concentrating on reusable rockets, could completely supplant space elevator development in the foreseeable future. According to him, the space elevator is attractive only because it offers cheaper ways to deliver cargo into orbit, and reusable rockets are being developed precisely to reduce the cost of this delivery.

Edwards blames the stagnation of the idea on the lack of real support for the project. “This is what projects look like that hundreds of people scattered around the world develop as a hobby. No serious progress will be made until there is real support and centralized control."

The situation with the development of the idea of a space elevator in Japan is different. The country is famous for its developments in the field of robotics, and Japanese physicist Sumio Iijima is considered a pioneer in the field of nanotubes. The idea of a space elevator is almost national here.

Japanese company Obayashi vows to deliver a working space elevator by 2050. The company's chief executive, Yoji Ishikawa, says they are working with private contractors and local universities to improve existing nanotube technology.

Ishikawa says that although the company understands the complexity of the project, they do not see any fundamental obstacles to its implementation. He also believes that the popularity of the idea of a space elevator in Japan is caused by the need to have some kind of national idea that unites people against the backdrop of the difficult economic situation of the last couple of decades.

Ishikawa is confident that although an idea of this scale can most likely only be realized through international cooperation, Japan could well become its locomotive due to the great popularity of the space elevator in the country.

Meanwhile, Canadian space and defense company Thoth Technology was awarded U.S. Patent No. 9,085,897 last summer for their version of a space elevator. More precisely, the concept involves the construction of a tower that retains its rigidity thanks to compressed gas.

The tower should deliver cargo to a height of 20 km, from where they will be launched into orbit using conventional rockets. This intermediate option, according to the company’s calculations, will save up to 30% of fuel compared to a rocket.

According to theoretical calculations, they seem to be a suitable material. If we assume their suitability for the manufacture of a cable, then the creation of a space elevator is a solvable engineering problem, although it requires the use of advanced developments and. NASA is already funding related developments at the American Institute for Scientific Research, including the development of a lift capable of moving independently along a cable. Presumably, this method in the future could be orders of magnitude cheaper than using launch vehicles.

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Subtitles

Design

The technology for weaving such fibers is still in its infancy.

According to some scientists, even carbon nanotubes will never be strong enough to make a space elevator cable.

In June 2013, engineers from Columbia University in the USA reported a new breakthrough: thanks to a new technology for producing graphene, it is possible to obtain sheets with a diagonal size of several tens of centimeters and a strength only 10% less than theoretical.