vignesh instrumentation

In celestial mechanics where we are dealing with planetary or stellar sized bodies, it is often the case that the mass of the secondary body is significant in relation to the mass of the primary, as with the Moon and Earth. In this case the size of the secondary cannot be ignored. The distance R is no longer negligible compared to r and, therefore, must be carried through the derivation. Equation (4.9) becomes
More commonly the equation is written in the equivalent form where a is the semi-major axis. The semi-major axis used in astronomy is always the primary-to-secondary distance, or the geocentric semi-major axis. For example, the Moon's mean geocentric distance from Earth (a) is 384,403 kilometers. On the other hand, the Moon's distance from the barycenter (r) is 379,732 km, with Earth's counter-orbit (R) taking up the difference of 4,671 km.

Kepler's second law of planetary motion must, of course, hold true for circular orbits. In such orbits both and r are constant so that equal areas are swept out in equal times by the line joining a planet and the sun. For elliptical orbits, however, both and r will vary with time. Let's now consider this case. Figure 4.5 shows a particle revolving around C along some arbitrary path. The area swept out by the radius vector in a short time interval t is shown shaded. This area, neglecting the small triangular region at the end, is one-half the base times the height or approximately r(rt)/2. This expression becomes more exact as t approaches zero, i.e. the small triangle goes to zero more rapidly than the large one. The rate at which area is being swept out instantaneously is therefore
For any given body moving under the influence of a central force, the value r² is constant. Let's now consider two points P₁ and P₂ in an orbit with radii r₁ and r₂, and velocities v₁ and v₂. Since the velocity is always tangent to the path, it can be seen that if is the angle between r and v, then
where vsin is the transverse component of v. Multiplying through by r, we have
or, for two points P₁ and P₂ on the orbital path
Note that at periapsis and apoapsis, = 90 degrees. Thus, letting P₁ and P₂ be these two points we get
Let's now look at the energy of the above particle at points P₁ and P₂. Conservation of energy states that the sum of the kinetic energy and the potential energy of a particle remains constant. The kinetic energy T of a particle is given by mv²/2 while the potential energy of gravity V is calculated by the equation -GMm/r. Applying conservation of energy we have
From equations (4.14) and (4.15) we obtain
Rearranging terms we get
Click here for example problem #4.4
Click here for example problem #4.5 The eccentricity e of an orbit is given by
Click here for example problem #4.6 If the semi-major axis a and the eccentricity e of an orbit are known, then the periapsis and apoapsis distances can be calculated by
Click here for example problem #4.7 Launch of a Space Vehicle
The launch of a satellite or space vehicle consists of a period of powered flight during which the vehicle is lifted above the Earth's atmosphere and accelerated to orbital velocity by a rocket, or launch vehicle. Powered flight concludes at burnout of the rocket's last stage at which time the vehicle begins its free flight. During free flight the space vehicle is assumed to be subjected only to the gravitational pull of the Earth. If the vehicle moves far from the Earth, its trajectory may be affected by the gravitational influence of the sun, moon, or another planet. A space vehicle's orbit may be determined from the position and the velocity of the vehicle at the beginning of its free flight. A vehicle's position and velocity can be described by the variables r, v, and , where r is the vehicle's distance from the center of the Earth, v is its velocity, and is the angle between the position and the velocity vectors, called the zenith angle (see Figure 4.7). If we let r₁, v₁, and ₁ be the initial (launch) values of r, v, and , then we may consider these as given quantities. If we let point P₂ represent the perigee, then equation (4.13) becomes
Substituting equation (4.23) into (4.15), we can obtain an equation for the perigee radius R_p.
Multiplying through by -R_p²/(r₁²v₁²) and rearranging, we get
Note that this is a simple quadratic equation in the ratio (R_p/r₁) and that 2GM /(r₁ × v₁²) is a nondimensional parameter of the orbit. Solving for (R_p/r₁) gives
Like any quadratic, the above equation yields two answers. The smaller of the two answers corresponds to R_p, the periapsis radius. The other root corresponds to the apoapsis radius, R_a. Please note that in practice spacecraft launches are usually terminated at either perigee or apogee, i.e. = 90. This condition results in the minimum use of propellant.
Click here for example problem #4.8 Equation (4.26) gives the values of R_p and R_a from which the eccentricity of the orbit can be calculated, however, it may be simpler to calculate the eccentricity e directly from the equation
Click here for example problem #4.9 To pin down a satellite's orbit in space, we need to know the angle , the true anomaly, from the periapsis point to the launch point. This angle is given by
Click here for example problem #4.10 In most calculations, the complement of the zenith angle is used, denoted by . This angle is called the flight-path angle, and is positive when the velocity vector is directed away from the primary as shown in Figure 4.8. When flight-path angle is used, equations (4.26) through (4.28) are rewritten as follows:
The semi-major axis is, of course, equal to (R_p+R_a)/2, though it may be easier to calculate it directly as follows:
Click here for example problem #4.11 If e is solved for directly using equation (4.27) or (4.30), and a is solved for using equation (4.32), R_p and R_a can be solved for simply using equations (4.21) and (4.22). Orbit Tilt, Rotation and Orientation Above we determined the size and shape of the orbit, but to determine the orientation of the orbit in space, we must know the latitude and longitude and the heading of the space vehicle at burnout. Figure 4.9 above illustrates the location of a space vehicle at engine burnout, or orbit insertion. is the azimuth heading measured in degrees clockwise from north, is the geocentric latitude (or declination) of the burnout point, is the angular distance between the ascending node and the burnout point measured in the equatorial plane, and is the angular distance between the ascending node and the burnout point measured in the orbital plane. ₁ and ₂ are the geographical longitudes of the ascending node and the burnout point at the instant of engine burnout. Figure 4.10 pictures the orbital elements, where i is the inclination, is the longitude at the ascending node, is the argument of periapsis, and is the true anomaly. If , , and ₂ are given, the other values can be calculated from the following relationships:
In equation (4.36), the value of is found using equation (4.28) or (4.31). If is positive, periapsis is west of the burnout point (as shown in Figure 4.10); if is negative, periapsis is east of the burnout point. The longitude of the ascending node, , is measured in celestial longitude, while ₁ is geographical longitude. The celestial longitude of the ascending node is equal to the local apparent sidereal time, in degrees, at longitude ₁ at the time of engine burnout. Sidereal time is defined as the hour angle of the vernal equinox at a specific locality and time; it has the same value as the right ascension of any celestial body that is crossing the local meridian at that same instant. At the moment when the vernal equinox crosses the local meridian, the local apparent sidereal time is 00:00. See this sidereal time calculator. Click here for example problem #4.12 Position in an Elliptical Orbit
Johannes Kepler was able to solve the problem of relating position in an orbit to the elapsed time, t-t_o, or conversely, how long it takes to go from one point in an orbit to another. To solve this, Kepler introduced the quantity M, called the mean anomaly, which is the fraction of an orbit period that has elapsed since perigee. The mean anomaly equals the true anomaly for a circular orbit. By definition,
where M_o is the mean anomaly at time t_o and n is the mean motion, or the average angular velocity, determined from the semi-major axis of the orbit as follows:
This solution will give the average position and velocity, but satellite orbits are elliptical with a radius constantly varying in orbit. Because the satellite's velocity depends on this varying radius, it changes as well. To resolve this problem we can define an intermediate variable E, called the eccentric anomaly, for elliptical orbits, which is given by
where is the true anomaly. Mean anomaly is a function of eccentric anomaly by the formula
For small eccentricities a good approximation of true anomaly can be obtained by the following formula (the error is of the order e³):
The preceding five equations can be used to (1) find the time it takes to go from one position in an orbit to another, or (2) find the position in an orbit after a specific period of time. When solving these equations it is important to work in radians rather than degrees, where 2 radians equals 360 degrees. Click here for example problem #4.13
Click here for example problem #4.14 At any time in its orbit, the magnitude of a spacecraft's position vector, i.e. its distance from the primary body, and its flight-path angle can be calculated from the following equations:
And the spacecraft's velocity is given by,
Click here for example problem #4.15 Orbit Perturbations
The orbital elements discussed at the beginning of this section provide an excellent reference for describing orbits, however there are other forces acting on a satellite that perturb it away from the nominal orbit. These perturbations, or variations in the orbital elements, can be classified based on how they affect the Keplerian elements. Secular variations represent a linear variation in the element, short-period variations are periodic in the element with a period less than the orbital period, and long-period variations are those with a period greater than the orbital period. Because secular variations have long-term effects on orbit prediction (the orbital elements affected continue to increase or decrease), they will be discussed here for Earth-orbiting satellites. Precise orbit determination requires that the periodic variations be included as well. Third-Body Perturbations The gravitational forces of the Sun and the Moon cause periodic variations in all of the orbital elements, but only the longitude of the ascending node, argument of perigee, and mean anomaly experience secular variations. These secular variations arise from a gyroscopic precession of the orbit about the ecliptic pole. The secular variation in mean anomaly is much smaller than the mean motion and has little effect on the orbit, however the secular variations in longitude of the ascending node and argument of perigee are important, especially for high-altitude orbits. For nearly circular orbits the equations for the secular rates of change resulting from the Sun and Moon are Longitude of the ascending node:
Argument of perigee:
where i is the orbit inclination, n is the number of orbit revolutions per day, and and are in degrees per day. These equations are only approximate; they neglect the variation caused by the changing orientation of the orbital plane with respect to both the Moon's orbital plane and the ecliptic plane.
Click here for example problem #4.16 Perturbations due to Non-spherical Earth When developing the two-body equations of motion, we assumed the Earth was a spherically symmetrical, homogeneous mass. In fact, the Earth is neither homogeneous nor spherical. The most dominant features are a bulge at the equator, a slight pear shape, and flattening at the poles. For a potential function of the Earth, we can find a satellite's acceleration by taking the gradient of the potential function. The most widely used form of the geopotential function depends on latitude and geopotential coefficients, J_n, called the zonal coefficients. The potential generated by the non-spherical Earth causes periodic variations in all the orbital elements. The dominant effects, however, are secular variations in longitude of the ascending node and argument of perigee because of the Earth's oblateness, represented by the J₂ term in the geopotential expansion. The rates of change of and due to J₂ are
where n is the mean motion in degrees/day, J₂ has the value 0.00108263, R_E is the Earth's equatorial radius, a is the semi-major axis in kilometers, i is the inclination, e is the eccentricity, and and are in degrees/day. For satellites in GEO and below, the J₂ perturbations dominate; for satellites above GEO the Sun and Moon perturbations dominate. Molniya orbits are designed so that the perturbations in argument of perigee are zero. This conditions occurs when the term 4-5sin²i is equal to zero or, that is, when the inclination is either 63.4 or 116.6 degrees. Click here for example problem #4.17 Perturbations from Atmospheric Drag Drag is the resistance offered by a gas or liquid to a body moving through it. A spacecraft is subjected to drag forces when moving through a planet's atmosphere. This drag is greatest during launch and reentry, however, even a space vehicle in low Earth orbit experiences some drag as it moves through the Earth's thin upper atmosphere. In time, the action of drag on a space vehicle will cause it to spiral back into the atmosphere, eventually to disintegrate or burn up. If a space vehicle comes within 120 to 160 km of the Earth's surface, atmospheric drag will bring it down in a few days, with final disintegration occurring at an altitude of about 80 km. Above approximately 600 km, on the other hand, drag is so weak that orbits usually last more than 10 years - beyond a satellite's operational lifetime. The deterioration of a spacecraft's orbit due to drag is called decay. The drag force F_D on a body acts in the opposite direction of the velocity vector and is given by the equation
where C_D is the drag coefficient, is the air density, v is the body's velocity, and A is the area of the body normal to the flow. The drag coefficient is dependent on the geometric form of the body and is generally determined by experiment. Earth orbiting satellites typically have very high drag coefficients in the range of about 2 to 4. Air density is given by the appendix Atmosphere Properties. The region above 90 km is the Earth's thermosphere where the absorption of extreme ultraviolet radiation from the Sun results in a very rapid increase in temperature with altitude. At approximately 200-250 km this temperature approaches a limiting value, the average value of which ranges between about 600 and 1,200 K over a typical solar cycle. Solar activity also has a significant affect on atmospheric density, with high solar activity resulting in high density. Below about 150 km the density is not strongly affected by solar activity; however, at satellite altitudes in the range of 500 to 800 km, the density variations between solar maximum and solar minimum are approximately two orders of magnitude. The large variations imply that satellites will decay more rapidly during periods of solar maxima and much more slowly during solar minima. For circular orbits we can approximate the changes in semi-major axis, period, and velocity per revolution using the following equations:
where a is the semi-major axis, P is the orbit period, and V, A and m are the satellite's velocity, area, and mass respectively. The term m/(C_DA), called the ballistic coefficient, is given as a constant for most satellites. Drag effects are strongest for satellites with low ballistic coefficients, this is, light vehicles with large frontal areas. A rough estimate of a satellite's lifetime, L, due to drag can be computed from
where H is the atmospheric density scale height. A substantially more accurate estimate (although still very approximate) can be obtained by integrating equation (4.53), taking into account the changes in atmospheric density with both altitude and solar activity. Click here for example problem #4.18 Perturbations from Solar Radiation Solar radiation pressure causes periodic variations in all of the orbital elements. The magnitude of the acceleration in m/s² arising from solar radiation pressure is
where A is the cross-sectional area of the satellite exposed to the Sun and m is the mass of the satellite in kilograms. For satellites below 800 km altitude, acceleration from atmospheric drag is greater than that from solar radiation pressure; above 800 km, acceleration from solar radiation pressure is greater. Orbit Maneuvers
At some point during the lifetime of most space vehicles or satellites, we must change one or more of the orbital elements. For example, we may need to transfer from an initial parking orbit to the final mission orbit, rendezvous with or intercept another spacecraft, or correct the orbital elements to adjust for the perturbations discussed in the previous section. Most frequently, we must change the orbit altitude, plane, or both. To change the orbit of a space vehicle, we have to change its velocity vector in magnitude or direction. Most propulsion systems operate for only a short time compared to the orbital period, thus we can treat the maneuver as an impulsive change in velocity while the position remains fixed. For this reason, any maneuver changing the orbit of a space vehicle must occur at a point where the old orbit intersects the new orbit. If the orbits do not intersect, we must use an intermediate orbit that intersects both. In this case, the total maneuver will require at least two propulsive burns. Orbit Altitude Changes The most common type of in-plane maneuver changes the size and energy of an orbit, usually from a low-altitude parking orbit to a higher-altitude mission orbit such as a geosynchronous orbit. Because the initial and final orbits do not intersect, the maneuver requires a transfer orbit. Figure 4.11 represents a Hohmann transfer orbit. In this case, the transfer orbit's ellipse is tangent to both the initial and final orbits at the transfer orbit's perigee and apogee respectively. The orbits are tangential, so the velocity vectors are collinear, and the Hohmann transfer represents the most fuel-efficient transfer between two circular, coplanar orbits. When transferring from a smaller orbit to a larger orbit, the change in velocity is applied in the direction of motion; when transferring from a larger orbit to a smaller, the change of velocity is opposite to the direction of motion. The total change in velocity required for the orbit transfer is the sum of the velocity changes at perigee and apogee of the transfer ellipse. Since the velocity vectors are collinear, the velocity changes are just the differences in magnitudes of the velocities in each orbit. If we know the initial and final orbits, r_A and r_B, we can calculate the total velocity change using the following equations:
Note that equations (4.59) and (4.60) are the same as equation (4.6), and equations (4.61) and (4.62) are the same as equation (4.45).
Click here for example problem #4.19 Ordinarily we want to transfer a space vehicle using the smallest amount of energy, which usually leads to using a Hohmann transfer orbit. However, sometimes we may need to transfer a satellite between orbits in less time than that required to complete the Hohmann transfer. Figure 4.12 shows a faster transfer called the One-Tangent Burn. In this instance the transfer orbit is tangential to the initial orbit. It intersects the final orbit at an angle equal to the flight path angle of the transfer orbit at the point of intersection. An infinite number of transfer orbits are tangential to the initial orbit and intersect the final orbit at some angle. Thus, we may choose the transfer orbit by specifying the size of the transfer orbit, the angular change of the transfer, or the time required to complete the transfer. We can then define the transfer orbit and calculate the required velocities. For example, we may specify the size of the transfer orbit, choosing any semi-major axis that is greater than the semi-major axis of the Hohmann transfer ellipse. Once we know the semi-major axis of the ellipse, a_tx, we can calculate the eccentricity, angular distance traveled in the transfer, the velocity change required for the transfer, and the time required to complete the transfer. We do this using equations (4.59) through (4.63) and (4.65) above, and the following equations:

Click here for example problem #4.20 Another option for changing the size of an orbit is to use electric propulsion to produce a constant low-thrust burn, which results in a spiral transfer. We can approximate the velocity change for this type of orbit transfer by
where the velocities are the circular velocities of the two orbits. Orbit Plane Changes To change the orientation of a satellite's orbital plane, typically the inclination, we must change the direction of the velocity vector. This maneuver requires a component of V to be perpendicular to the orbital plane and, therefore, perpendicular to the initial velocity vector. If the size of the orbit remains constant, the maneuver is called a simple plane change. We can find the required change in velocity by using the law of cosines. For the case in which V_f is equal to V_i, this expression reduces to
where V_i is the velocity before and after the burn, and is the angle change required.
Click here for example problem #4.21 From equation (4.73) we see that if the angular change is equal to 60 degrees, the required change in velocity is equal to the current velocity. Plane changes are very expensive in terms of the required change in velocity and resulting propellant consumption. To minimize this, we should change the plane at a point where the velocity of the satellite is a minimum: at apogee for an elliptical orbit. In some cases, it may even be cheaper to boost the satellite into a higher orbit, change the orbit plane at apogee, and return the satellite to its original orbit. Typically, orbital transfers require changes in both the size and the plane of the orbit, such as transferring from an inclined parking orbit at low altitude to a zero-inclination orbit at geosynchronous altitude. We can do this transfer in two steps: a Hohmann transfer to change the size of the orbit and a simple plane change to make the orbit equatorial. A more efficient method (less total change in velocity) would be to combine the plane change with the tangential burn at apogee of the transfer orbit. As we must change both the magnitude and direction of the velocity vector, we can find the required change in velocity using the law of cosines,
where V_i is the initial velocity, V_f is the final velocity, and is the angle change required. Note that equation (4.74) is in the same form as equation (4.69). Click here for example problem #4.22 As can be seen from equation (4.74), a small plane change can be combined with an altitude change for almost no cost in V or propellant. Consequently, in practice, geosynchronous transfer is done with a small plane change at perigee and most of the plane change at apogee. Another option is to complete the maneuver using three burns. The first burn is a coplanar maneuver placing the satellite into a transfer orbit with an apogee much higher than the final orbit. When the satellite reaches apogee of the transfer orbit, a combined plane change maneuver is done. This places the satellite in a second transfer orbit that is coplanar with the final orbit and has a perigee altitude equal to the altitude of the final orbit. Finally, when the satellite reaches perigee of the second transfer orbit, another coplanar maneuver places the satellite into the final orbit. This three-burn maneuver may save propellant, but the propellant savings comes at the expense of the total time required to complete the maneuver. When a plane change is used to modify inclination only, the magnitude of the angle change is simply the difference between the initial and final inclinations. In this case, the initial and final orbits share the same ascending and descending nodes. The plane change maneuver takes places when the space vehicle passes through one of these two nodes. In some instances, however, a plane change is used to alter an orbit's longitude of ascending node in addition to the inclination. An example might be a maneuver to correct out-of-plane errors to make the orbits of two space vehicles coplanar in preparation for a rendezvous. If the orbital elements of the initial and final orbits are known, the plane change angle is determined by the vector dot product. If i_i and _i are the inclination and longitude of ascending node of the initial orbit, and i_f and _f are the inclination and longitude of ascending node of the final orbit, then the angle between the orbital planes, , is given by
Click here for example problem #4.23 The plane change maneuver takes place at one of two nodes where the initial and final orbits intersect. The latitude and longitude of these nodes are determined by the vector cross product. The position of one of the two nodes is given by
Knowing the position of one node, the second node is simply
Click here for example problem #4.24 Orbit Rendezvous Orbital transfer becomes more complicated when the object is to rendezvous with or intercept another object in space: both the interceptor and the target must arrive at the rendezvous point at the same time. This precision demands a phasing orbit to accomplish the maneuver. A phasing orbit is any orbit that results in the interceptor achieving the desired geometry relative to the target to initiate a Hohmann transfer. If the initial and final orbits are circular, coplanar, and of different sizes, then the phasing orbit is simply the initial interceptor orbit. The interceptor remains in the initial orbit until the relative motion between the interceptor and target results in the desired geometry. At that point, we would inject the interceptor into a Hohmann transfer orbit. Launch Windows Similar to the rendezvous problem is the launch-window problem, or determining the appropriate time to launch from the surface of the Earth into the desired orbital plane. Because the orbital plane is fixed in inertial space, the launch window is the time when the launch site on the surface of the Earth rotates through the orbital plane. The time of the launch depends on the launch site's latitude and longitude and the satellite orbit's inclination and longitude of ascending node. Orbit Maintenance Once in their mission orbits, many satellites need no additional orbit adjustment. On the other hand, mission requirements may demand that we maneuver the satellite to correct the orbital elements when perturbing forces have changed them. Two particular cases of note are satellites with repeating ground tracks and geostationary satellites. After the mission of a satellite is complete, several options exist, depending on the orbit. We may allow low-altitude orbits to decay and reenter the atmosphere or use a velocity change to speed up the process. We may also boost satellites at all altitudes into benign orbits to reduce the probability of collision with active payloads, especially at synchronous altitudes. V Budget To an orbit designer, a space mission is a series of different orbits. For example, a satellite might be released in a low-Earth parking orbit, transferred to some mission orbit, go through a series of resphasings or alternate mission orbits, and then move to some final orbit at the end of its useful life. Each of these orbit changes requires energy. The V budget is traditionally used to account for this energy. It sums all the velocity changes required throughout the space mission life. In a broad sense the V budget represents the cost for each mission orbit scenario. The Hyperbolic Orbit
The discussion thus far has focused on the elliptical orbit, which will result whenever a spacecraft has insufficient velocity to escape the gravity of its primary. There is a velocity, called the escape velocity, V_esc, such that if the spacecraft is launched with an initial velocity greater than V_esc, it will travel away from the planet and never return. To achieve escape velocity we must give the spacecraft enough kinetic energy to overcome all of the negative gravitational potential energy. Thus, if m is the mass of the spacecraft, M is the mass of the planet, and r is the radial distance between the spacecraft and planet, the potential energy is -GmM /r. The kinetic energy of the spacecraft, when it is launched, is mv²/2. We thus have
which is independent of the mass of the spacecraft.
Click here for example problem #4.25

A space vehicle that has exceeded the escape velocity of a planet will travel a hyperbolic path relative to the planet. The hyperbola is an unusual and interesting conic section because it has two branches. The arms of a hyperbola are asymptotic to two intersecting straight line (the asymptotes). If we consider the left-hand focus, f, as the prime focus (where the center of our gravitating body is located), then only the left branch of the hyperbola represents the possible orbit. If, instead, we assume a force of repulsion between our satellite and the body located at f (such as the force between two like-charged electric particles), then the right-hand branch represents the orbit. The parameters a, b and c are labeled in Figure 4.14. We can see that c² = a²+ b² for the hyperbola. The eccentricity is,
The angle between the asymptotes, which represents the angle through which the path of a space vehicle is turned by its encounter with a planet, is labeled . This turning angle is related to the geometry of the hyperbola as follows:
If we let equal the angle between the periapsis vector and the departure asymptote, i.e. the true anomaly at infinity, we have
If we know the radius, r, velocity, v, and flight path angle, , of a point on the orbit (see Figure 4.15), we can calculate the eccentricity and semi-major axis using equations (4.30) and (4.32) as previously presented. Note that the semi-major axis of a hyperbola is negative. The true anomaly corresponding to known valves of r, v and can be calculated using equation (4.31), however special care must be taken to assure the angle is placed in the correct quadrant. It may be easier to first calculate e and a, and then calculate true anomaly using equation (4.43), rearranged as follows:
Whenever is positive, should be taken as positive; whenever is negative, should be taken as negative. The impact parameter, b, is the distance of closest approach that would result between a spacecraft and planet if the spacecraft trajectory was undeflected by gravity. The impact parameter is,
Closet approach occurs at periapsis, where the radius distance, r_o, is equal to
p is a geometrical constant of the conic called the parameter or semi-latus rectum, and is equal to
Click here for example problem #4.26 At any known true anomaly, the magnitude of a spacecraft's radius vector, its flight-path angle, and its velocity can be calculated using equations (4.43), (4.44) and (4.45). Click here for example problem #4.27 Early we introduced the variable eccentric anomaly and its use in deriving the time of flight in an elliptical orbit. In a similar manner, the analytical derivation of the hyperbolic time of flight, using the hyperbolic eccentric anomaly, F, can be derived as follows:
where,
Whenever is positive, F should be taken as positive; whenever is negative, F should be taken as negative. Click here for example problem #4.28 Hyperbolic Excess Velocity If you give a space vehicle exactly escape velocity, it will just barely escape the gravitational field, which means that its velocity will be approaching zero as its distance from the force center approaches infinity. If, on the other hand, we give our vehicle more than escape velocity at a point near Earth, we would expect the velocity at a great distance from Earth to be approaching some finite constant value. This residual velocity the vehicle would have left over even at infinity is called hyperbolic excess velocity. We can calculate this velocity from the energy equation written for two points on the hyperbolic escape trajectory – a point near Earth called the burnout point and a point an infinite distance from Earth where the velocity will be the hyperbolic excess velocity, v_∞. Solving for v_∞ we obtain
Note that if v_∞ = 0 (as it is on a parabolic trajectory), the burnout velocity, v_bo, becomes simply the escape velocity. Click here for example problem #4.29 It is, or course, absurd to talk about a space vehicle "reaching infinity" and in this sense it is meaningless to talk about escaping a gravitational field completely. It is a fact, however, that once a space vehicle is a great distance from Earth, for all practical purposes it has escaped. In other words, it has already slowed down to very nearly its hyperbolic excess velocity. It is convenient to define a sphere around every gravitational body and say that when a probe crosses the edge of this sphere of influence it has escaped. Although it is difficult to get agreement on exactly where the sphere of influence should be drawn, the concept is convenient and is widely used, especially in lunar and interplanetary trajectories. For most purposes, the radius of the sphere of influence for a planet can be calculated as follows:
where D_sp is the distance between the Sun and the planet, M_p is the mass of the planet, and M_s is the mass of the Sun. Equation (4.89) is also valid for calculating a moon's sphere of influence, where the moon is substituted for the planet and the planet for the Sun.

What is the difference between a Single acting and double acting valve?

2013-01-30T08:40:00.002-08:00

Normally a control valve is refer to by it's fail position. This means "what position will the valve move to should the supply air or control signal to the valve falls away". This is important to safe guard the process at various places so some valves will be fail open and some fail close. In order to have valve as a fail open or close the valve the actuator have to be spring loaded. So by having the spring on top or bottom of the actuator piston, will determine if it will be a FO or FC valve. This kind of valve is also called single action since it will only have one output from its positioner to either the top or bottom of the actuator. The positioner on the valve is also setup as a single acting positioner since it will only give a single action to the actuator, the reverse action will be done by the spring. The problem with this setup is that it is possible that the process might be so strong or the pressure so high (during a blow down or ESD shutdown in the plant) that the spring might in certain instances be to week to push the valve into the fail position quick enough, due to the back pressure from the process and can cause damage to the plant or even a explosion. To make sure that the valve will go to the fail position we install a double action positioner with two outputs. One goes to the top of the actuator and one to the bottom. This is also very helpful to do very accurate and stable control on a high flow line since the pressure from the position do the actual control and not spring control one way and positioner control the other way as in single acting control valves. It is also solving the problem that the valve will now be forced into the fail position by the spring as well as the positioner supply pressure during a emergency. In shutdown valves (open/close ESDV's) the same is true and sometime at critical and high pressure points we use hydraulics instead of pneumatics as the double acting agent to make sure the valve will close during a emergency. So to summarize the double acting action in ESD and control valve is just there to make sure the valve will do what it was designed for. Call it a extra fail safe if you want. In theory not needed since a single acting valve should do the trick just as well,but in practice you are at time very glad you did it especially if you look at the kind of pressures the valves are working on. With those kind of flows and pressures you don't want to leave anything to chance

Nucleonic level transmitter

2012-02-22T22:04:00.000-08:00

Process Level Measurement

Traditional sensors such as displacers and differential pressure
transmitters are making way for new technology and in particular
two-wire radar level transmitters.

We look at some of the advances in different level technologies
and highlight why pulse radar is rapidly becoming the preferred
choice for process level measurement.

Capacitive / Admittance level transmitters
Capacitive level transmitters have been used on a variety of liquid
and solid level applications for many years. If the process is
understood and does not vary, and if a number of basic rules
are followed, then capacitive level sensors can prove to be
reliable and cost effective.

In the past, applications with very viscous and conductive products
have proved to be problematic. However, recent advances such
as "phase selective admittance" processing within the transmitter
oscillator have allowed automatic compensation for product
build-up on the probe. This technology is available with two wire
4 to 20 mA / HART and Profibus PA directly from the transmitter head.

"Phase selective admittance" electronics processing
is designed to ignore conductive build-up.
Two-wire transmitters are available with 4-20mA/HART
and Profibus PA.

Hydrostatic level transmitters
Differential pressure transmitters have been used widely in vented
storage tanks as well as in pressurised vessels. The use of pressure
to measure level has obvious limitations particularly when the
density changes between different products. Many older designs
have liquid filled chemical seals and capillaries which do not lend
themselves to accurate level measurement.

Temperature variations on the liquid fill cause measurement
inaccuracies and the process diaphragm is often thin and prone
to mechanical damage.

There have been significant new developments in hydrostatic
sensors designed specifically for level measurement. Mechanically
robust and chemically resistant Sapphire-Ceramic measurement
cells are available with flush, hygienic process connections.These sensors do not have a liquid fill and can be fitted at the very
bottom of dished vessels. They have no "dead volume" that can
cause blockages or hygiene problems within food and pharmaceutical
processes. High accuracy is achieved with these direct mounted
devices.

"Sapphire-Ceramic" hydrostatic level transmitter with flush, hygienic process connection. The design is mechanically robust and abrasion resistant with no "dead" pockets. The sensor can be mounted very low in the vessel. It is ideal for food and pharmaceutical applications.

Ultrasonic level transmitters
The benefits of non-contact level transmitters were first realised
with the advent of ultrasonic level transmitters back in the 1970's.

Ultrasonics are suitable for water based liquids and for solids
applications where the effects of pneumatic filling are not too extreme.

In recent years, transducer technology has been improved and
more importantly echo processing software is more capable than
ever before. It is fair to say that ultrasonic level sensors have been
pushed to their physical limits. Even the most powerful ultrasonic
transducers are affected by vapours, gas composition, excess dust,
steam and condensation as well as filling noise and air movement.
Obviously, ultrasonics do not work in a vacuum.

Having said this, ultrasonics still provide a relatively low cost,
non-contact level transmitter with ranges from less than one metre
to 70 metres and above. Ultrasonic transmitters are available with
a variety of analogue and digital output options.

Non-contact ultrasonic level transmitters are used on liquids and solids with measuring ranges up to 70 metres.
Longer ranges generally require larger, lower frequency transducers.

Nucleonic or Radiometric level transmitters
Nucleonic (or radiometric) level transmitters are one of the only
truly non-invasive forms of level measurement. They are able to
measure liquids or solids through the wall of stainless steel or
other metal vessels.

This measurement technique, the differential adsorption of Gamma
radiation from a radioactive isotope, such as Caesium Cs137 or
Cobalt 60, is used to detect the level in the vessel.

This radiation source is specially shielded so that the radiation
is only directed towards the product through the vessel wall.
The radiation source is mounted near the maximum level on
one side of the vessel while radiation detectors are mounted
down the opposite side of the vessel. The amount of radiation
blocked is proportional to the level.

"FibreFlex" is a unique scintillation detector
It uses a special fibre-optic bundle of 7 metres or more that can be wrapped around the contours of a vessel.
In-head smart electronics complete this innovative product.

This measurement technique is very capable and is applied in
some of the most challenging process level applications in the
petrochemical, chemical and offshore industries as well as in
heavy solids handling applications typical in the quarrying and
cement industries.

Ohmart-VEGA have continued to develop nucleonic measurement
into the 21st century. One innovation is the revolutionary "FiberFlex"
flexible scintillation detector. Standard ion chamber and Geiger-Mueller
tube defectors are heavy with low sensitivity and normal scintillation
tube detectors are rigid and limited in length.

"FiberFlex" utilizes a special fibre-optic bundle that can follow the
contours of the process vessel. It is light in weight, easy to transport
and install and it can be longer than standard detectors at over
7 metres.

Advances in detector technology include in-head HART / 4 to 20 mA
transmiller, Foundation Fieldbus transmitters for scintillation
detectors including "FiberFlex".

The nucleonic technique can also be used for interface level
measurement and for density measurement.

Guided microwave level transmitters
The guided microwave level measurement technique is a variation
of radar. Also known as Time Domain Reflectometry, or TDR,
extremely short microwave pulses are transmitted down the
outside of a cable or rod and reflected off the surface of the product
that is being measured. The measurement possibilities include
level measurement in liquids and solids plus interface measurement
between low dielectric non-conductive liquids and conductive liquids.

In solids applications, guided microwave sensors look similar to
cable capacitive probes. However, they have the advantage of
being unaffected by changing product characteristics and they
can be calibrated without filling the vessel. Also, they are
independent of the internal structure of the silo or hopper and
many different powders and granular products can be measured.

Guided microwave or TDR level transmitters for different
powder and granular solids.
Guided microwave or 'radar on a rope'. It can be calibrated without filling the silo and is independent of changes in the characteristics of the product being measured.

Different designs are used on liquids applications. This includes
dual rod and concentric tube/rod designs. The advantage of these
designs is that the electrical field around the probe is concentrated
in the tube or between the rods and there is little external influence
from the vessel structure. However, these designs are susceptible
to build up of product between the rods or within the tube. This
can lead to measurement problems with false echoes caused by
the build-up. They are better suited for clean liquids.

Two-wire, loop powered guided microwave level transmitters for
liquids and solids are available with HART /4 to 20 mA and Profibus PA.

Radar level measurement
The benefits of radar as a level measurement technique are
compelling and clear. Radar provides a non-contact sensor that
is virtually unaffected by the following process conditions

* Changes in process temperature,
* Changes in process pressure
* Vacuum
* Variations in the gas and vapour composition.
* Air movement between the sensor and the product surface
* Density of the product
* Conductivity of the product
* Dielectric constant of the product

Pulse radar in particular, presents additional capabilities and
benefits in process vessel level applications.

Using time of flight measurement, pulse radar does not need
the expensive and power consuming processors that enable
the alternative FM-CW (frequency modulated continuous wave)
radar technique to function.

There are no Fast Fourier Transform (FF1) algorithms to calculate.
Instead, the echoes derived from a pulse radar are discrete and
separated in time. All of the processing is dedicated to echo
analysis alone.

This means that pulse radar is better equipped to handle multiple
echoes and false echoes that are common in process vessels.
Some FM-CW radar transmitters cannot cope with multiple echoes
that occur in simple horizontal cylindrical vessels due to the
parabolic effect of the vessel top. This does not trouble pulse radar.
The low power consumption requirements of pulse have ensured
that a very capable iwo-wire intrinsically safe radar has been realised.

The averaging of the pulse technique reduces the noise curve to
allow smaller echoes to be detected. Well designed circuits
containing good quality electronic components ensure that pulse
radar can detect echoes over a wide dynamic range of about 80 dB.
This can make the difference between reliable and unreliable
measurement.

VEGA produced the world's very first iwo-wire, loop powered
radar level transmitter in the summer of 1997. The VEGAPULS 50
series set the standard in performance for process radar.
Overnight, two-wire technology reduced the unit cost of radar
sensors and ensured that radar became an affordable first choice
level measurement technique.

These two-wire radar level transmitters have been proved on
some of the most arduous applications from agitated chemical
and pharmaceutical process reactors to solids applications
including fly ash and cement clinker silos.

The still unrivalled, "ECHOFOX" echo processing software is
at the heart of this success.

5.8 GHz radar
Example: VEGAPULS 54 6" horn is optimum size.

Choice of radar frequency
Process radar level transmitters operate at microwave frequencies
between 5.8 GHz and 26 GHz. Manufacturers have chosen
frequencies for different reasons rangin from licensing considerations,
the availability of microwave components and perceived technical
advantages.

There are arguments extolling the virtues of high frequency radar,
low frequency radar and every frequency radar in between.
In reality, no single radar frequency is ideal for all radar level
applications.

It is a question of physics. Which frequency is best for the particular
process conditions or vessel characteristics Low frequency: 58 GHz
5.8 GHz radar is ideally suited to difficult process vessel applications
such as chemical reactors and solids level applications. The lower
frequency is less affected by condensation, build-up, agitation and
foam.

In general terms, 5.8 GHz is more "forgiving" than higher frequencies.
In addition, 5.8 GHz is capable of measuring liquid ammonia where
higher frequencies signals are damped.

5.8 0Hz radar
Example: VEGAPULS 53 PTFE dielectric rod antenna.

High frequency: 26 GHz
26 GHz is ideally suited for bulk storage tanks, horizontal cylinders,
mixing tanks, small vessels such as receivers and also bypass
tubes and stilling tubes.

The high frequency radar has better focusing for a given antenna
size. As a consequence, a higher accuracy can be achieved and
smaller horn antennas can be used. The focusing of high frequency
radar avoids many false echoes and there is very little influence
from standard vessel nozzles.

When combined with the pulse technique, there is virtually no
near-range dead band thus ensuring the maximum vessel volume
is usable.

26 GHz radar
Example: VEGAPULS 42
Smallest horn antenna radar with
a diameter of 40mm.

Non-contact process radar has developed at a rapid pace.
Antenna designs for more and more arduous process environments;
echo processing software that can cope with the false echoes
from the internal structure of a vessel; a choice of radar frequencies
to ensure that the correct radar is chosen for the myriad of possible
applications; very capable and affordable two-wire loop powered
pulse radar transmitters.

All of these factors lead to the conclusion that the future of process
level measurement lies in non-contact pulse radar.

Waveguide radar
With very low dielectric constant liquids, such as butane (dielectric
constant of 1 .4) and propane (dielectric constant of 1 .6), a stilling
tube is required to enable measurement with radar.

Essentially, the stilling tube is a wave-guide that concentrates the
microwaves and ensures that the signal amplitude is high enough
for reliable measurement. A high frequency radar design for
liquefied gases and other clean liquid hydrocarbons uses an
integral stilling tube. The design is optimised for maximum accuracy
and has the benefit of being pre-calibrated.

Radar applications
Two-wire, intrinsically safe 5.8 GHz
radar on a small process reactor vessel
application.

Ultimate chemical resistance
A special enamel (glass) coated metallic horn antenna is aimed
at the chemical and pharmaceutical industries. This has all the
benefits of a horn antenna with excellent materials compatibility.
The enamel coating is the same material as the glass lined vessels
for which they are designed.

The only other "wetted" material is the PTFE waveguide inside
the enamel coated steel horn. The 5.8 GHz frequency makes this
radar ideal for chemical reactor applications.

High frequency pulse radar measuring
hot oil in a stirred process vessel.

High temperature and pressure
Pressurized process vessels in the chemical and petrochemical
industries are required to withstand constant high temperatures
at continuous high pressures.

These arduous process conditions are met by using radar with
a chemically and thermally stable ceramic waveguide. The whole
assembly is laser welded to ensure that the transmitter is gas
tight and that differential thermal expansion is negligible.
The electronics housing are mechanically isolated from the high
process temperatures by a temperature extension tube.
The design is capable of withstanding 400 °C at a constant
pressure of 160 Bar.
___________________________________________________

The author of this article, Peter Devine has published a book called
"Radar level measurement - the user's guide".

The guide is a full and comprehensive publication in full colour hardback A5 format covering the following:

Part I
1. History of radar
2. Physics of radar
3. Types of radar.
Part II
4. Radar level measurement
5. Radar antennas
6. Radar level installations
a. Mechanical installation
b. Electrical installation
Part III
7. Other level techniques:
Amongst others, DP transmitters, Capactive transmitters, Electromechanical transmitters and Ultrasonic.
8. Applications:
Covering applications in the following industries: Brewing & Distilling,
Cement, Chemical, Food, Metals, Offshore, Pharmaceutical, Power and Water & Waste water industries.
Part IV
9. VEGA radar
Two wire loop powered radar
Vegapuls 50 - technical specification
Vegapuls 40 - technical specification
Vegapuls 40 & 50 options
Echofox software

The book also has additional appendices, which cover the following:
Glossary of terms, Radar power & radar power density, Dielectric constants Symbols, Photograph acknowledgements.

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