Lunarpedia - User contributions [en]

Luna Project

2009-04-14T06:01:12Z

Publius:

{{Org Stub}}

The Luna Project is a cooperative, volunteer effort to organize knowledge and expertise needed to build a city on the lunar surface. The organization is currently active and has a highly ambitious timetable for lunar return.

Current activities of the Project include attempting to develop support within the broader space interest community through attendance and promotions at science-fiction conventions and industry events, a [http://www.lunarcc.org/materials/index.html#bookpgm giveaway program] for space related books, distribution of publications such as the [http://lunarcc.org/materials/index.html#lpub periodical '''Luna!'''] and other avenues of approach ; and attempting to build involvement and organize effort through means of communication including a [http://www.lunarcc.org/toplevel/contact.html#listserv mailing list] and a [http://forum.lunarcc.org/ Web forum].

==External Links==
[http://lunarcc.org/ Luna Project website]

Talk:Lunar Transfer Trajectories

2008-11-09T06:14:31Z

Publius: discussion of "Lunar transfer trajectories" page

I am doing a series of MATLAB calculations on an idealized case of "ballistic transfer trajectories", the simplification putting Luna in circular rigid-body motion about Terra with a period equal to the sidereal month, at a radius equal to the mean barycentric distance. I did this before with a slightly different set of assumptions. The objective is an exhaustive search of first-revolution trajectories, at 1 m/s (starting geocentric velocity) by 1 degree (initial position with respect to Terra-Luna radius vector) resolution. I expect to search the velocity range from 10 000 to 12 000 m/s, with a maximum flight-time of 240 hours. Then I will pick out some nice ones to plot graphs of.
<br/>-- [[User:Publius|publius]] 06:14, 9 November 2008 (UTC)

Talk:Long Endurance Rovers

2008-11-09T04:07:31Z

Publius: talk page for "Long Endurance Rovers"

I don't have time to edit this page right now, so I'm going to summarize what I see as some of the problems.

; Organization
: The various sub-topics seem to be spread around at random
; Content
: No mention is made of the Russian "Lunokhods" (generally considered teleoperated rovers, although they were provided with manual controls). Lunokhod 1 operated for almost a year, & Lunokhod 2 for about 6 months. Their landing sites were in relatively high latitudes, where the daytime temperature peaks are lower, & they hibernated during the night with a very clever mechanism (heat radiators were located under a "lid" which was opened when the interior got warm & shut when it got cold).
: The cooling mechanism described for APOLLO is wrong. There was an "oxygen purge system" in the PLSS (backpack), but it was purely a backup in case of a failure of the primary system, which didn't actually happen. Suit cooling was provided by a flash-evaporator using water. The LEM was cooled primarily by radiation, with an evaporator for use under higher heat loads. (As a side comment, if we are going to waste anything, oxygen is not a bad choice, being the most abundant single element in Luna & one of the easiest to produce.)
: Some of the items just don't belong in this article, e.g. the description of sun-shield berms (which work better in middle and high latitudes anyway).
: The description of the "parabolic reflectors" is much too specific in terms of defining the shape, & in any case a diagram or sketch would be preferable. Excessive specificity strikes again elsewhere in the description -- it's not clear why the heat-circulating fluid should be "a mixture of water and ethylene glycol" rather than, say, alcohol, sodium-potassium alloy, or any of a number of other suitable liquids, or why it should be a liquid rather than a gas (e.g. helium), or a a phase-change fluid ("heat-pipe" radiators have proven very effective). "Sintered brick shelters" & "pumped storage of high-pressure oxygen", although possible approaches for night shelter & power storage respectively, are certainly not the only possible ones, & need evaluation.
<BR/>-- [[User:Publius|publius]] 04:07, 9 November 2008 (UTC)

User:Publius

2008-11-01T04:22:59Z

Publius: Description of user

==A.K.A.==

Christopher Carson is better known in some circles as ''publius'', an alias he uses extensively, not out of desire for anonymity, but because "Christophers" & "Carsons" are so common that his own name is not a good identifier. Much to his chagrin, he is the driving force behind the [[Luna Project]].

Lunar Transfer Trajectories

2008-10-31T18:29:59Z

Publius:

==Introduction==

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

==Ballistic Trajectories==

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

===Low-energy Trajectories===

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown<ref>V. A. Egorov, "Certain Problems of Moon Flight Dynamics", ap. ''The Russian Literature of Satellites, Part 1''. New York: International Physical Index, 1958.</ref> that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the [[Lagrangian point]] L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on [[weak boundary capture]] using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the [[Interplanetary Transport Network]], but in general they are similar to the Jacobi-integral trajectory.

===First-revolution Elliptical Trajectories===

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

===Parabolic Trajectories===

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

===Hyperbolic Trajectories===

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

==Non-Ballistic Trajectories==

===Constant-thrust Trajectories===

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.<ref>Maxwell Hunter, ''Thrust Into Space''. New York: Holt, Rinehart, & Winston, 1966. See pp.148 et seq.</ref> The [[Deep Space One]] spacecraft provides a good example of a "classical" low-thrust mission. The more recent [[Smart-1]] lunar probe, using a Hall-effect thruster, combined elements of constant-thrust, multiple-impulse, & low-energy trajectories to transfer from geosynchronous transfer orbit (having been "piggy-backed" on a communications satellite) into lunar orbit over the course of more than a year.

===Multiple-impulse Trajectories===

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The [[Chandrayaan-1]] probe is currently employing this type of maneuver to transfer into lunar orbit.

==References==
<references/>

[[category:Celestial mechanics]]

Lunar Transfer Trajectories

2008-10-31T18:26:27Z

Publius:

==Introduction==

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

==Ballistic Trajectories==

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

===Low-energy Trajectories===

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown<ref>V. A. Egorov, "Certain Problems of Moon Flight Dynamics", ap. ''The Russian Literature of Satellites, Part 1''. New York: International Physical Index, 1958.</ref> that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the [[Lagrangian point]] L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on [[weak boundary capture]] using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the [[Interplanetary Transport Network]], but in general they are similar to the Jacobi-integral trajectory.

===First-revolution Elliptical Trajectories===

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

===Parabolic Trajectories===

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

===Hyperbolic Trajectories===

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

==Non-Ballistic Trajectories==

===Constant-thrust Trajectories===

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.<ref>Maxwell Hunter, ''Thrust Into Space''. New York: Holt, Rinehart, & Winston, 1966. See pp.148 et seq.</ref> The [[Deep Space One]] spacecraft provides a good example of a "classical" low-thrust mission. The more recent [[SMART-1]] lunar probe, using a Hall-effect thruster, combined elements of constant-thrust, multiple-impulse, & low-energy trajectories to transfer from geosynchronous transfer orbit (having been "piggy-backed" on a communications satellite) into lunar orbit over the course of more than a year.

===Multiple-impulse Trajectories===

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The [[Chandrayaan-1]] probe is currently employing this type of maneuver to transfer into lunar orbit.

==References==
<references/>

[[category:Celestial mechanics]]

Lunar Transfer Trajectories

2008-10-31T06:57:08Z

Publius: /* Low-energy Trajectories */

==Introduction==

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

==Ballistic Trajectories==

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

===Low-energy Trajectories===

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown[1] that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the [[Lagrangian point]] L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on [[weak boundary capture]] using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the [[Interplanetary Transport Network]], but in general they are similar to the Jacobi-integral trajectory.

===First-revolution Elliptical Trajectories===

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

===Parabolic Trajectories===

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

===Hyperbolic Trajectories===

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

==Non-Ballistic Trajectories==

===Constant-thrust Trajectories===

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.[2] An important recent example is the [[SMART-1]] lunar probe, which employed a Hall-effect thruster to transfer from an eccentric terrestrial orbit into lunar orbit over the course of more than a year.

===Multiple-impulse Trajectories===

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The [[Chandrayaan-1]] probe is currently employing this type of maneuver to transfer into lunar orbit.

==References==
#V. A. Egorov, "Certain Problems of Moon Flight Dynamics", ap. ''The Russian Literature of Satellites, Part 1''. New York: International Physical Index, 1958.
#Maxwell Hunter, ''Thrust Into Space''. New York: Holt, Rinehart, & Winston, 1966. See pp.148 et seq.

[[category:Celestial mechanics]]

Lunar Transfer Trajectories

2008-10-31T06:56:26Z

Publius: /* Low-energy Trajectories */

==Introduction==

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

==Ballistic Trajectories==

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

===Low-energy Trajectories===

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown[1] that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the [[Lagrangian point]] L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on [[Weak boundary capture]] using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the [[Interplanetary Transport Network]], but in general they are similar to the Jacobi-integral trajectory.

===First-revolution Elliptical Trajectories===

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

===Parabolic Trajectories===

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

===Hyperbolic Trajectories===

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

==Non-Ballistic Trajectories==

===Constant-thrust Trajectories===

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.[2] An important recent example is the [[SMART-1]] lunar probe, which employed a Hall-effect thruster to transfer from an eccentric terrestrial orbit into lunar orbit over the course of more than a year.

===Multiple-impulse Trajectories===

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The [[Chandrayaan-1]] probe is currently employing this type of maneuver to transfer into lunar orbit.

==References==
#V. A. Egorov, "Certain Problems of Moon Flight Dynamics", ap. ''The Russian Literature of Satellites, Part 1''. New York: International Physical Index, 1958.
#Maxwell Hunter, ''Thrust Into Space''. New York: Holt, Rinehart, & Winston, 1966. See pp.148 et seq.

[[category:Celestial mechanics]]

Lunar Transfer Trajectories

2008-10-31T06:55:00Z

Publius: A summary of trajectories from Terra to Luna

==Introduction==

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

==Ballistic Trajectories==

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

===Low-energy Trajectories===

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown[1] that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the [[Lagrangian point]] L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on [[Weak boundary capture]] using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the [[Interplanetary Superhighway]], but in general they are similar to the Jacobi-integral trajectory.

===First-revolution Elliptical Trajectories===

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

===Parabolic Trajectories===

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

===Hyperbolic Trajectories===

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

==Non-Ballistic Trajectories==

===Constant-thrust Trajectories===

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.[2] An important recent example is the [[SMART-1]] lunar probe, which employed a Hall-effect thruster to transfer from an eccentric terrestrial orbit into lunar orbit over the course of more than a year.

===Multiple-impulse Trajectories===

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The [[Chandrayaan-1]] probe is currently employing this type of maneuver to transfer into lunar orbit.

==References==
#V. A. Egorov, "Certain Problems of Moon Flight Dynamics", ap. ''The Russian Literature of Satellites, Part 1''. New York: International Physical Index, 1958.
#Maxwell Hunter, ''Thrust Into Space''. New York: Holt, Rinehart, & Winston, 1966. See pp.148 et seq.

[[category:Celestial mechanics]]