Determination of Lagrange Points06-03-2009 05:38

The Lagrangian points are locations in space in the vicinity of two orbiting masses where the gravitational forces and the orbital motion balance each other to form a point at which a third body of negligible mass would be stationary relative to the two bodies.

There are a number of web resources describing the Lagrangian points, some of them offering a very good introduction and description of the 5 Lagrangian points, while others insist on the techical aspects of the problem. Nonetheless, in the following article we'll try to add to the non-mathematical and intuitive descriptions available on the web (like, for instance, theESA page related to the 5 lagrangian pointsor theWikipedia article) a detailed-mathematical view of the problem, starting from the technical defintion which states that the Lagrangian points are the stationary solutions of the circular restricted three-body problem.

If we consider the restricted 3 body problem:

Considering an equilibrium point P we have the condition:

Then the resultant force is:

If we introduce the center of mass of a system like being a specific point where the systems mass behaves as the system is concentrated in that specific point we will have the following relation:

For the particular Sun-Earth system the barycenter is located as following:

In the casewe have(449 km for the Sun-Earth system)Now lets introduce in the system the Coriolis force and the centrifugal force.Coriolis force:

Centrifugal force:

Then the resultant force is:

Correspondingly, the generalized potential is:

The first term from the formula abovedetermines the position of the equilibrium points, and the second onedetermines the stability of motion about the equilibrium points.If we consider a Cartesian coordinates system around the center of mass of the system Sun-Earth as in the following picture:

Considering all above we obtain the resultant force:

In order to determine the static equilibrium points we put the condition

Then the resultant force became:

From the conditionwe will find the position of the equilibrium points.

In particular, if we considerwe can find the first 3 Lagrange points (located in the line Sun-Earth).

Now, in order to simplify the calculation we consider:

Lets replace in the above formula the fact, that.

Now we have basically three possibilities (the fourth case is practically impossible):

Correspondingly to the above cases we will have three equations:(1)(2)(3)Or if we want to express the equations in a general form we can write:

Then the three particular cases become:(1)(2)(3)We have now to solve these 3 equations. We can do this either by using an approximation considering the particularity of the system Earth-Sunor we can solve it numerically using the Newton method.Lets first consider the approximationand try to find a series which approximate the real solution.We then have in the first equation:

If we note

For the second equation we have:

For the third equation:

In the second case can be proven that the three equations have all just one real solution the other 4 being complex. The real solutions can be found being in the interval:Using a numerical program and replacing the constants in the particular case of the system Earth-Moon-Sun we will have: M1=1.98892E30;//mass of Sun [kg] M2=5.9742E24;//mass of Earth [kg] M3=7.36E22;//mass of Moon [kg] R=1.4959610E8;//1 AU=R [km]; al=(M2+M3)/(M1+M2+M3);//alpha be=M1/(M1+M2+M3);//betaThen the roots of the three equations (calculated with a precision of 1e-14) are: x1=-0.0100113196805154 x2=0.010078587510165 x3=-0.0100332069481937So the physical coordinates of the first three Lagrange points of the system Earth-Moon-Sun are: L1x=148097990.737622 L1y=0 L2x=151103362.50271 L2y=0 L3x=148094716.487738 L3y=0expressed in km from the center of mass of the system.In the particular case of the L2 point which will focus on we found it at 1507717.38502939 km from Earth on the line which lies the Sun and the Earth.Lets find now the solutions for the other 2 Lagrange points.

We have the resultant force:

Considering the two directions (paralleland perpendicular) we obtain the projection on the perpendicular direction:

Ifand

Lets consider now the projection on the parallel direction:Replacing the previous solution

we have:

From the condition

So the two solutions are.Then the last two Lagrange points are located as:

Physically the solutions are found replacing the constants for the system Earth-Moon-Sunexpressed in kilometers from the center of mass of the system.Lissajous orbits around L2 pointWe have to find the general solution of the system of equations around L2 point:

with:

The general solution of the above system of equations can be proven to be of the form:

where,

whereis the solution of the equation which gives us the L2 point.

Based on the initial conditions the coefficients can be found as:

where:

We can simplify the system of equations:

If we put the supplementary conditionwe obtain a Lissajous orbit in the liniarized restricted three body problem:

with.Ifwe obtain a Halo orbit around L2.The following pictures show a typical Lissajous orbit with the conditions:

The following pictures show a general Lissajous orbit:

As can be seen, the exponential part of the equations results in a very fast grow of the orbit and, as consequence, the satellite will escape from the desired orbit. In conclusion the satellite should be kept in a liniarised Lissajous orbit such that the exponent is null.

Short term versus precise long term calculation of the proper motion of a star06-03-2009 06:07

The concept of proper motion in astronomy is defined as the apparent movement of a star on the celestial sphere, usually measured as seconds of arc per year. It is due both to the actual relative motions of the sun and the star through space. The proper motion reflects only transverse motion (the component of motion across the line of sight to the star); it does not include the component of motion toward or away from the sun. The most distant stars show the least proper motion. The average proper motion of the stars that can be seen with the naked eye is 0.1'' per year.

The proper motion of a star is its apparent angular movement per year on the celestial sphere. It is a combination of its actual motion through space and its motion relative to the solar system. Most stars are so distant that the proper motion is almost negligible.The full space motion of a star is the combination of this proper motion (which is a quantity measured by the Hipparcos satellite) with the "radial velocity" of the star, along the line of sight. The radial velocity of stars or galaxies, are usually measured from the Doppler shifts in their spectra: this quantity was not measured by Hipparcos. Stars dont have a fix position in space, they move around so over the time the sky changes as well. Short term calculations apply when the star moves only a small distance or angle. For most of the stars, considering their slow motion, the short term refers to something less than 10000 years.In order to start computing the proper motion of a star, one should consider the following elements from the Hipparcos catalogue: - as the right ascension (in decimal hours) - as the declination (in decimal degrees) - the proper motion in right ascension (in arc seconds per year) - the proper motion in declination (in arc seconds per year)For short term calculations, it will be assumed that the proper motion in both right ascension and declination are constant. Most of the time, considering the small angular motion, this is a fair enough assumption. Then the new values of right ascension and declination are found as: t= 0+ t t= 0+ tThe only thing that should be carried on is to keep the original units used when reading the stars data from the catalogue.This simple approach works well for time changes up to about 10000 years past or future compared with the time when the catalogue data have been produced. However the calculation losses the accuracy for longer times, since the stars distance from Earth changes appreciably and so its proper motion changes, invalidating this simple approach. As a general rule, if the star moves by more than about 30 degrees- roughly the length of the Big Dipper or the height of Orion, these calculations are braked down. After a few hundred thousand years, about any star will have moved enough to invalidate this simple calculation. Therefore a more complex model for calculating the proper motion can be put in the scene.In order to accurately calculate the stellar motion, the stars motion in three dimensions should be known. The three conventional space velocity components are: Rthe stars radial velocity the proper motion in right ascension the proper motion in declination dthe stars distance in parsecsIf the radial velocity is missing, a null value can be used in all the following calculations. If the catalogue gives the distances in light years, the distance should be divided by 3262 in order to get parsecs. If the catalogue gives a parallaxpa, it can be converted to distance by using the parallax formulapa=1/d(parallaxes less than about 0.01 arc seconds in older catalogues or 0.001 arc seconds in the Hipparcos catalogue are usually considered poor, and should be used carefully)...It should be considered that both components of proper motion are in the same units (seconds of arc per year) - if not they should be converted accordingly. Then they should be converted from these angular units, to linear velocities- the two components of the transverse velocityTof the star. To make these velocities consistent with the radial velocity , they should be converted into km/s. TRA= (arcsec/year) d 4740 (km/s) TDEC= (arcsec/year) d 4740 (km/s) These velocities are used to calculate the change in position of a star over time. However these velocities as calculated are hard to work with. First their orientation in space will vary from star to star.TDEC, for instance, points towards the celestial poles if the star has a declination of zero, but points 90 degrees away from the poles if the star is at the pole. It is generally easier to transform these velocities to Cartesian velocities with components along some consistent set of axes. If we use the coordinate system as: +x axis towards = 0degrees, = 6hours (vernal equinox) +y axis towards = 0degrees, = 6hours +z axis towards = 90degrees (north celestial pole)The three Cartesian velocities can be calculated in terms of the three velocitiesR,TRAandTDEC:

The Cartesian velocities retain the original unit (that is km/s), but for interstellar motions it is more appropriate to express distances in terms of parsecs and times in terms of years rather than seconds. Then the natural speed unit will turn to be expressed in parsecs per year. A conversion from km/s to pc/year is obtained by dividing by 977780.If we express the three Cartesian coordinates for a star at time t = 0 (present time): We can calculate the new positions at time t considering that for the time frames we are interested in (a few thousand years to a million years) the stellar motion is pretty much linear - stars are too far apart for the gravity to curve their paths appreciably. Now it can go back to equatorial coordinates (right ascension and declination) using the following formulas: The result can be converted from degrees to hours by dividing the result to 15.Over a few thousand years, a stars motion generally doesnt change its brightness very much, but as these calculations are highly accurate over millions of years, it can be seen that many stars will get measurably brighter or fainter. If we introduce in the following equations the apparent magnitude V as seen from Earth: The calculation is identical to the one for calculating the brightness of a star as seen from a different reference point in space- a given change in distance to a star, whether its observer that moves or the star, yields the same effect on brightness.The following objects are the ten highest proper motion stars contained in the Hipparcos Catalogue (the 61 Cygni binary is seen at this resolution only as a single object):Name of star or regionRADec

Barnard's star269.44.6

Kapteyn's star77.8-45.0

Groombridge 1830178.237.7

Lacaille 9352346.4-35.8

CD -37 154921.3-37.3

HIP 67593207.723.7

61 Cygni A & 61 Cygni B316.738.7

Lalande 21185165.835.9

epsilon Indi330.8-56.8

Attitude determination-Summary of the attitude sensors06-03-2009 04:02This article intends to give to the reader an overview of the state of the art attitude sensors used for the modern missions.

Attitude determination uses a combination of sensors and mathematical models to collect vector components in the body and inertial reference frames, typically in the form of a quaternion, Euler angles or rotation matrix.The main sensors available on satellite area are: Earth sensorsThere are different earth sensors available on the market like horizon crossing indicators and horizon scanners. The sensors use the difference of the infrared light emitted by Earth and the one emitted by the deep space in order to detect the horizon, and then they return the vector to the approximate center of the Earth. Sun sensorThe sun sensors are available in two forms: coarse sun sensors (used usually to determine the attitude of the satellite in a safe mode when the accuracy is not an important issue) and fine sun sensors (used whenever a big accuracy is requested). Star sensorThey are the most accurate sensors used for attitude determination at the present. It is important that they deliver a full attitude determination, meaning that they dont need another vector measurement since the measurement of the stars in the field of view already provides an attitude solution.The process consists in taking a picture of the sky, comparing this picture with a star map stored on board of the spacecraft and based on some specific algorithms identifying the stars found and generating an attitude solution.Future notes will give the details of the specific star sensors topics. MagnetometerThey measure the vector and the intensity of the magnetic field at the current point in orbit of the satellite. As the spacecraft is carrying a database with an accurate magnetic field model, then if the orbital position has been identified, the vector can be used for attitude determination. However, the quality of the vector measurement depends on the quality of the magnetic field model stored in the onboard computer as well as on the current events from ionosphere (like magnetic storms). GyroThey are used to measure the angular velocities and not directly the attitude of the satellite.Attitude control-Summary of the attitude actuators06-03-2009 04:24This article intends to be a summary of the attitude actuators for spacecrafts either active or passive. It will introduce the reader to the basic attitude control hardware in use for the modern platforms.

The attitude control of a spacecraft can be considered being either actively controlled (meaning that a controller calculates necessary control torques and acting on the satellite to adjust its attitude to a desired position) or passively controlled (meaning that the satellite uses external torques that occurs due to its interaction with the environment and thus they cannot be avoided, in this case the disturbances being used for forcing the attitude of the satellite).In general, the active control assures 3 axes stabilization, while the passive control gives the opportunity for 1 axis stabilization. Some disadvantageous points of the passive actuators are that the pointing accuracy is pretty bad and also that the natural damping is very small meaning that additional energy dissipation devices need to be installed on board of the spacecraft.A summary of the active actuators include: ThrustersThe torques generated by the thrusters is considered as external torques since the angular momentum of the entire satellite changes. The accuracy of the attitude control depends on the minimum impulse of the type of thruster used.Depending on the size of the satellite and taking in consideration the complexity of this solution, different types of thrusters are normally being used: gas jets, ion jets or even nuclear propulsion.Gas jets produce thrust by a collective acceleration of propellant molecules, with the energy coming from either a chemical reaction or thermodynamic expansion.Gas jets are classified as hot gas when the energy is derived from a chemical reaction or cold gas when it is derived from the latent heat of a phase change, or from the work of compression if no phase change is involved. Hot gas jets generally produce a higher thrust level (>5 N) and a greater total impulse or time integral of the force. Cold gas systems operate more consistently, particularly when the system is operated in a pulsed mode, because there is no chemical reaction which must reach steady state. The lower thrust levels (500 N) can be obtained, but the complexity of a two component system is justified only when these thrust levels are required. Monopropellant systems use a catalyst or less frequently, high temperature to promote decomposition of a single component, which is commonly hydrazine (N2H4) or hydrogen peroxide (H2O2). Hydrazine with catalytic decomposition is the most frequently used hot gas monopropellant system on spacecraft. The problem of consistency manifests in two ways. First, the thrust is bellow nominal for the initial few seconds of firing because the reaction rate is bellow the steady state value until the catalyst bed reaches operating temperature. Second, the thrust profile, or time dependence of thrust, changes as a function of total thruster firing time; this is significant when a long series of short pulses is executed, because the thrust profile for the later pulses will differ from that for the earlier pulses.The propellant supply required for jets is the major limitation on their use; a fuel budget is an important part of mission planning for any system using jets. Other considerations are the overall weight of the system and the need to position thrusters where the exhaust will not impinge on the spacecraft. The latter consideration is especially important when hydrazine is used, because the exhaust contains ammonia, which is corrosive.In more distant orbits, jets are the only practical means of interchanging momentum with the environment. High thrust or total impulse requirements may indicate a hot gas system. Otherwise the cold gas system may be favored because hydrazine freezes at about 0 deg C and may require heaters if lower temperatures will be encountered during the mission. Specific components may affect the relative system reliability; for example hydrazine systems use tank diaphragms to separate the propellant from the pressurizing agent and also require a catalyst or heater to initiate decomposition; cold gas systems may have a pressure regulator between the tank and the thruster.Ion jets accelerate individual ionized molecules electro-dynamically, with the energy ultimately coming from solar cells or self containing electric generators.The first successful orbital test of the ion engine was made in 1968, and the engine was used to keep a satellite in geo-synchronous orbit. The test demonstrated that ion propulsion was possible in space, and that the electrical system did not interfere with any of the other systems on the spacecraft.Ion jets are primarily used in applications that do not require large amounts of thrust, such as satellite control. The thrust produced by a typical ion engine is on the order of millinewtons, and thus cannot yet be used as a primary propulsion system for launching any spacecraft from the earths surface. However, the spacecraft utilizing the ion jet engine can be launched via chemical rocket, and then sufficient thrust can be developed in the frictionless atmosphere of space.While chemical rocket engines produce thrust by burning large amounts of fuel and expelling hot gases, ion jets do so by expelling highly accelerated ions of inert gases. This greatly reduces the amount of required propellant, and removes limitations found in conventional rocket engines caused by the high thermal stresses of combustion. Angular momentum storage and exchange devicesThese devices consist of a spinning wheel, either the orientation or the spin rate being changed. In the absence of external torques, the angular momentum of the entire satellite does not change, but whenever the rotation rate or the rotation axis of such a device is changed, the satellite will experience a torque in such a way that the angular momentum of the entire satellite system is constant.On the same manner these devices can be used to compensate the external disturbance torques (which can cause a change in the angular momentum of the satellite) by producing their own internal torques.The main advantage of these devices is that they assure very high accuracies, and their main disadvantage is that they have to be de-saturated using an actuator that generates external torques in the case when a maximum rotation rate is reached. One should also consider that such devices have big power consumptions and masses.There are three types of such devices: momentum wheels, reaction wheels and fly wheels. MagnetorquersMost of the time magnetorquers are used as coils but more generally any conducting device can be used to perform this function. They are often used in combination with angular momentum storage and exchange devices, or they are used to de-saturate the wheels without using propellant.The physical principle consists in generating a magnetic dipole moment in a desired direction, moment which will interact with the Earths magnetic field and thus external torques appear. However the use of such a device is limited to the low Earth orbits where the Earths magnetic field strength has usable values and should take in consideration that the generation of torques can be done just for the one perpendicular to the magnetic field vector. Considering this cosine dependency, should be very easily intuited the main disadvantage of this method- named that the absolute torque produced is very small.A summary of the passive actuators include: Gravity gradientThis method is based on the fact that the gravitational force decreases with the square of the distance. By building the satellite in a certain configuration, parts of the satellites which are closer from the Earths center are subject to a bigger force than the ones which are further away and in consequence this effect can be used to produce torques that adjust the satellite to a certain position. Magnetic dipoleThis method depends highly on the strength of the magnetic field in a specific point of spacecrafts orbit and this is the reason why it can be used only for the low Earth orbits where the values of the magnetic field strength are big enough. The method is based on installation on board of the satellite of a strong constant magnetic dipole or a permanent magnet, which will interact with the Earths magnetic field vector causing an adjustment of the two axes. Aerodynamic stabilizationThe aerodynamics of the satellite (influenced by the geometry of the shape) can be used in flight to adjust the position with respect to the flight direction (based on the forces caused by the atmospheric drag).However this method can only apply where the atmospheric drag is big enough (i.e. a low Earth orbit) and only for a short period of time, as this will lower the altitude and will shorten the decay period. Solar radiationThey can be used to generate torque on the spacecraft based on the solar radiation pressure created by the Sun. The orientation is then an axis that points to the Sun.