SensorsWeek-9

Compiled from Fundamentals of Spacecraft Attitude by F. Landis Markley John L. Crassidis

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artes.esa.int

Introduction• Attitude determination uses a combination of sensors and

mathematical models to collect vector components in the:– body and – inertial reference frames.

• These components are used in one of several differentalgorithms to determine the attitude, typically in the form of a quaternion, Euler angles, or a rotation matrix.

• It takes at least two vectors to estimate the attitude. • For example, an attitude determination system might use a

sun vector, s and a magnetic field vector m. • A sun sensor measures the components of s in the body

frame, sb, while a mathematical model of the Sun's apparent motion relative to the spacecraft is used to determine the components in the inertial frame, si.

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• Similarly, a magnetometer measures the components of min the body frame, mb, while a mathematical model of the Earth's magnetic field relative to the spacecraft is used to determine the components in the inertial frame, mi.

• An attitude determination algorithm is then used to find a rotation matrix Rbi such that:

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ibi

b

ibi

b

mRm

sRs

• The attitude determination analyst needs to understand:• how various sensors measure the body-frame

components, • how mathematical models are used to determine the

inertial-frame components, and • how standard attitude determination algorithms are used

to estimate Rbi.

Introduction

Underdetermined or Overdetermined

• In the previous section we make claim that at least two vectors are required to determine the attitude.

• Recall that it takes three independent parameters to determine the attitude, and that a unit vector is actually only two parameters because of the unit vector constraint.

• Therefore we require three scalars to determine the attitude.• Thus the requirement is for more than one and less than two vector

measurements.• The attitude determination is thus unique in that:

– one measurement is not enough, i.e., the problem is underdetermined, and

– two measurements is too many, i.e., the problem is overdetermined.

• The primary implication of this observation is that all attitude determination algorithms are really attitude estimation algorithms.

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Attitude Measurements• There are two basic classes of attitude sensors.

– The first class makes absolute measurements, – whereas the second class makes relative measurements.

• Absolute measurement sensors are based on the fact that knowing the position of a spacecraft in its orbit makes it possible to compute the vector directions, with respect to an inertial frame, of certain astronomical objects, and of the force lines of the Earth's magnetic field.

• Absolute measurement sensors measure these directions with respect to a spacecraft or body fixed reference frame, and by comparing the measurements with the known reference directions in an inertial reference frame, are able to determine (at leastapproximately) the relative orientation of the body frame with respect to the inertial frame.

• Absolute measurements are used in the static attitude determination algorithms developed in this chapter.

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• Relative measurement sensors belong to the class of gyroscopic instruments, including:– The rate gyro and– The integrating gyro.

• Classically, these instruments have been implemented as spinning disks mounted on gimbals; however, modern technology has brought such marvels as:– ring laser gyros, – fiber optic gyros, and – hemispherical resonator gyros.

• Relative measurement sensors are used in the dynamic attitude determination algorithms.

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Attitude Measurements

thetacticalbusiness.com

Sensors and Actutors• The history of attitude sensor development has emphasized

increased resolution and accuracy as well as decreased size, weight, and power (often abbreviated as SWaP).

• Actuator technologies have also been scaled down to be appropriate for microsatellites and Cubesats.

• We begin with a brief introduction to redundancy considerations, and then consider some specific sensors and actuators.

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Actuator Attitude System

Sensors

Input Output

Redundancy• The space environment is stressful, and failures of ACS components have

sometimes led to the degradation or premature termination of space missions.

• A requirement for many missions is the ability to survive the failure of any one component (single point), without any loss of capability.

• This is often accomplished by providing redundant components. • A cold backup, some designs leave the redundant equipment unpowered

until a failure of the operating unit occurs.• If a primary unit fails in a cold backup configuration, there is some delay

from the time that the backup component is powered on until it is available for use.

• A warm backup configuration, in which the redundant device is powered on but not used, allows a quicker recovery in the event of failure of the primary component.

• A hot backup configuration, with the redundant component being used all the time, is often preferred for actuators such as reaction wheels (movie)or CMGs (movie), where the increased control authority is useful.

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• Redundant components can be connected in:– a block-redundant or – a cross strapped fashion.

• This is illustrated schematically below, where sensors A and B might be a star tracker and a gyro, actuators A and B could be two reaction wheels, and the numerical subscripts represent redundant components.

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Redundancy

• The block-redundant system has two entirely separate strings;– a single-string system would just be the upper half of the figure. A

failure in the active string of a block-redundant system causes a switch to all the components of another string.

– A failure in the cross-strapped system would generally result in replacing only the failed unit with its backup.

• It can be seen that the block-redundant and cross-strappedconfigurations are both single-fault-tolerant.

• The block-redundant system would only tolerate a second fault if it were in the same string as the first fault.

• We can see that cross-strapping usually leads to a more robustsystem, because it can accommodate a greater range of multiple failures.

• There are instances, however, where cross-strapping can reduce reliability by allowing a fault in one component to propagate through the system.

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Redundancy

• Cross-strapping increases system– complexity and – cost.

• The increased expense is especially prominent in the testing phase, where it is desirable to test all the paths through the control system.

• It can be seen from the figure that the number of paths in a block-redundant system increases linearly with the degree of redundancy, but the number of paths increases exponentially in a fully cross-strapped system.

• It is often cost-effective to build a system with limited redundancy and/or partial cross strapping.

• A careful reliability analysis must be performed as part of the design process, to assess the degree of redundancy and cross-strapping needed to provide the desired probability of completing the mission successfully.

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Redundancy

• Hardware redundancy is not the only option for protecting against single-point failures.

• Instead, it is often possible to provide the function of a failed component by using an entirely different component or set of components.

• This often involves extra computation, and is referred to as analytic redundancy.

• One example is the provision for attitude determination of the Tropical Rainfall Measuring Mission (TRMM) spacecraft using gyros, Sun sensors, and a three-axis magnetometer inplace of the horizon sensor.

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Redundancy

www.spaceflightinsider.com

Sensors• The performance of a

spacecraft control system is limited by the:– performance of its sensors and – actuators.

• This chapter discusses the types of sensors that are used in spacecraft control systems.

• Many of the sensors are used for determining the attitude or attitude rates of the spacecraft.

• Others are used for determining the relative orientation or position ofcomponents on a spacecraft.

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directory.eoportal.org

Types of Sensors• Next Table lists the classes of sensors used in spacecraft.

• Within each class, there may be many types of sensors.

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Comparision of Current Attitude Sensors

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Star Trackers

• Beginning around 1990, they were superseded by solid state star trackers that track many stars simultaneously.

• They autonomously match the tracked stars with stars in an internal catalog to compute the star tracker’s attitude with respect to a celestial reference frame.

• A typical tracker has an update rate between 0.5 and 10 Hz, a mass of about 3 kg and a power requirement on the order of 10 W.

• It provides accuracy of a few arc seconds (10/3600) in the boresight pointingdirection, with larger errors for rotation about the boresight.

• Video1 StarTracker1:

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www.sodern.com

• A star tracker is basically a digital camera with a focal plane populated by either CCD(charge-coupled device) or CMOS(complementary metal-oxide semiconductor) pixels (picture elements).

• CCDs have lower noise, but CMOS has several advantages.

• It is the same technology used for microprocessors, so the pixels can include some data processing capabilities on the focal plane itself.

• Sensors taking advantage of this capability are known as active pixel sensors (APS).

• CMOS is more resistant to radiation damage than CCDs, and also provides the capability of reading out different pixels at different rates, which is not feasible with CCDs.

• Video2 Startracker2:

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Star Trackers

www.ericchesak.com

Star Tracker Focal Plane• Figure shows the geometry of a star

tracker, which is basically the geometry of a pinhole camera.

• The x, y, and z axes constitute a right-handed coordinate system with its origin at the vertex of the optical system and its z axis along theoptical axis, the tracker’s boresight.

• The focal plane is a distance f , the focal length of the optics, behind the vertex.

• The optics are slightly defocused so a star image covers several pixels.

• This enables the location of the centroid of a star image, computed as the “center of mass” of the photoelectrons in an nxn block of pixels, to be determined to an accuracy of a fraction of a pixel.

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• A centroiding algorithm measures the star locations in the optical detector.

• The fine optical control loop uses the star locations to determine the piezo stage commands to stabilize the science star image.

• The centroids are also used for satellite attitude determination. • Attitude determination is based on matching stars from the camera image

to the star catalogue. • Several sources of noise are introduced in the process of converting an

image to an attitude measurement including:– detector noise, – time delay, and – star catalogue errors.

• Errors in the centroiding algorithm along with detector noise will produce errors in the measured centroid location.

• Centroid errors can also cause improper matches to the star catalogue leading to attitude determination error.

• Therefore, it is important to utilize an accurate centroiding algorithm robust to detector noise.

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Star Tracker Centroiding

Centroiding Algorithm• Centroiding is a fundamental process in star pattern identification.• This is done so as to correctly locate the star centre in the star image

frame.• The image processing of star images are done to digitize the values

obtained for CCD and then reduce to a small observable set of values. • These are important for the further processing of the image.• The algorithm operates on the image and detects the bright spots in the

image.• But light is usually spread across an array of the CCD and it is difficult to

locate the exact center of the star.• This is mainly due to the spreading and defocusing of the light while

travelling through the optics of the camera.• The magnitude of the star and the spreading of the light determines the

size of the arrays that must be used in the Centroiding algorithm.• Usually it is 3x3 to 15x15.• The modern star centroiding algorithm helps to find the star image center

upto a precision on 1/10 of a pixel or even better.• The image centroid is thus important since it affects the accuracy of the

measurement and hence the whole process.

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• We will focuses star light from the nearby 3x3 pixels and finds the exact star center up to a pixel accuracy of 1/10.

• The star center is found out by detecting pixels which have a threshold value greater than a certain pre-defined specific value.

• The pixel should also have a value greater than is eight neighbouring pixels so as to ensure that the centroiding is done at the right star center.

• The value is defined during calibration of the camera.• Once the star is detected the surrounding values of the center pixels are

read and centroid is calculated from the formula given below.• The centroiding algorithm outputs centroid offset values (Xc,Yc) that

represents the offset of the calculated star location from the center of the star pixels of the 3x3 array.

• The centroid Xc and Yc can be found out by summing up the product of intensities and pixels locations given below:

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Centroiding Algorithm

3

1

3

1

3

1

3

1

.

i j

i j

cX

ji,I

iji,I

3

1

3

1

3

1

3

1

.

i j

i j

cY

ji,I

jji,I

• Example 0:

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Centroiding Algorithm

Star Triad Construction• An algorithm was created to analyze all

identified stars to build a star triad or multiple star triads with stars that meet the specified criteria.

• A triad is made up of;– a central star and – two secondary stars.

• The triad is only constructed if the secondary stars lie within a certain distance range from the central star to make sure they are not too close or too far away from the central star.

• Once a triad is constructed, the three angular distances between the stars are calculated.

• Multiple triads can be constructed for a single image to increase the probability that a triad will be successfully identified when compared to the star catalogue

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• Normally the triads are selected in a specific order.

– The first star selected is the brightest star in the image and,

– Then the second star is the second brighest star and,

– The third star is the third brightest star.

• The stars that make up the triad are numbered in the following way to beconsistent with the star comparison catalogue developed by Quine:

• Star 1 – The central star

• Star 2 – The star that forms the anticlockwise edge of the triad rotatedabout star 1

• Star 3 - The star that forms the clockwise edge of the triad

• The angular separation distances between the three stars are ordered in thefollowing way to be consistent with the star comparison catalogue:

– Star 1 - Star 2

– Star 1 – Star 3

– Star 2 – Star 3

Star Triad Construction

• The focal plane has a (u, v) coordinate system whose origin can be at its center or at one of its corners.

• The center of the focal plane, at the point where the z axis pierces it, is designated by (u0, v0).

• The unit vector s from the spacecraft to a star can be computed from the focal plane coordinates of the centroid of its image as:

• It is conventional to define:

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Star Tracker Frame

Example 1:

𝛼 ≡ 𝑡𝑎𝑛−1 Τ𝑠2 𝑠3𝛽 ≡ 𝑡𝑎𝑛−1 Τ𝑠1 𝑠3

u= 𝑢0 + 𝑓 Τ𝑠1 𝑠3=𝑢0 + 𝑓 tan 𝛼v= 𝑣0 + 𝑓 Τ𝑠2 𝑠3=𝑢0 + 𝑓 tan 𝛽

Star Tracker Modes of Operation• A star tracker has two modes of operation:

– tracking mode and– initial attitude acquisition.

• We will first discuss the more straightforward tracking mode, in which the tracker is following several stars that have already been matched with catalogued stars.

• After a fixed integration time, the star tracker reads out the number of accumulated photoelectrons in the pixels in regions of interest (ROI) around the expected positions of the tracked stars.

• The location of each ROI is based on the star’s position at the time of previous readout and the estimated attitude motion of the spacecraft in the intervening time, and its size depends on the accuracy of the attitude knowledge.

• The brightest pixels in each ROI are identified, and the appropriate nxn block of pixels is used to compute the centroid of each star.

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• If a tracked star moves out of the field of view (FOV), the tracker searches for another star, preferably well separated from the other tracked stars.

• The a priori knowledge of the approximate spacecraft attitude makes this search relatively easy.

• Initial attitude acquisition mode is more picturesquely known as lost-in-space mode.

• In this case the tracker searches the entire FOV for the brightest clusters of pixels, and computes at least three centroids.

• The arc length separation between these stars, their brightness, and some other computed properties are used to match them with entries in the star catalog.

• This can be accomplished in a few seconds using sophisticated algorithms for pattern matching and for rapidly searching thecatalog.

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Star Tracker Modes of Operation

Field of View, Resolution, Update Rate• The resolution of the star tracker depends on:

– the number of pixels,

– the size of the FOV, and the accuracy of the centroiding.

• We will consider a square focal plane of size NpixelsxNpixels

typical values being 512x512 or 1024x1024.

• Assuming that the focal plane assembly is centered on the optical axis, it images a spherical quadrilateral on the celestial sphere bounded by the four great circles with, α=+-αmax andβ=+-βmax.

• The area of this FOV on the sphere is given by spherical geometry as:

Ω𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 ≈ 2𝛼𝑚𝑎𝑥 2𝛽𝑚𝑎𝑥 steradian

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• The approximation is for small angles, and it is quite good, having an error of only 1% for αmax=βmax=100.

• Optical distortions often make it desirable to ignore stars in the corners of the FOV, reducing the useful area for αmax=βmax to that of a small circle of radius αmax,

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Field of View, Resolution, Update Rate

www.learnopencv.com

Ω𝑐𝑖𝑟𝑐𝑙𝑒 ≈ 𝜋 𝛼𝑚𝑎𝑥2

• Each pixel of a square focal plane subtends an angle 2βmax/Npixels in the small angle approximation, so the resolution of the tracker is 2κcentβmax/Npixels.

• Higher resolution can be obtained by:– decreasing the size of the FOV, – increasing the number of pixels in the focal plane, or

– improving the centroiding efficiency (κcent).

• If the physical size of a pixel and the field of view are held constant, adding pixels requires a larger focal plane and thus a proportionally larger focal length, increasing the weight of the optics.

• Pixel sizes have historically decreased, allowing more pixels in asmaller focal plane.

• If Nstars are tracked simultaneously, averaging of random errors reduces.

• In the attitude estimate about the two axes perpendicular to the boresight by a factor of (Nstars)

-1/2 , with resulting accuracy:

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Field of View, Resolution, Update Rate

Example 2:

• The rotation angle around the tracker’s boresight, often called roll, cannot be determined with equal accuracy.

• Its estimation requires some separation between the tracked stars to provide a lever arm.

• This error is independent of the size of the FOV.

• The error in the rotation about the boresight as:

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Field of View, Resolution, Update Rate(cont)

Example 3:

• We see that a large-FOV star tracker and a small-FOV tracker with the same focal plane will produce equally accurate roll attitude estimates, but the tracker with the smaller FOV will provide better measurements of the cross-axis attitude.

• A star catalog is the database where the algorithm searches to find a match for the star pattern.

• The star catalog contains data such as Right Ascension, Declination, Magnitude and the Triangular feature.

• The star catalog can be used in different ways to detect the star pattern either by utilizing:– the stellar coordinates, or

– stellar magnitudes

– or both.

• The Hipparcos catalog lists a little more than 118000 stars with 1 to 3 milli arc-sec level astronomy to magnitude 11.

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Star Catalogs

• Hipparcos contain:

– StarID

– RA, Dec: The star’s right ascension and declination for epoch 2000.0.

– Proper Name: Such as Orion or Sirius.

– Distance: The star’s distance in parsecs.

• 1 parsecs=3.262 light years

– X-Y-Z: Cartesian coordinates of the star in ECI frame

– VX,VY,VZ: The cartesian velocity components of the star.

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Star Catalogs

https://www.cosmos.esa.int/web/hipparcos/home

– Mag: Visual magnitude

– The units used to describe brightness of astronomical objects.

– The smaller the numerical value, the brighter the object.

– The human eye can detect stars to 6th or 7th magnitude on a dark, clear night far from city lights; in suburbs or cities, stars may only be visible to mag 2 or 3 or 4, due to light pollution.

– The brightest star, Sirius, shines at visual magnitude -1.5.

– Jupiter can get about as bright as visual magnitude -3 and Venus as bright as -4.

– The full moon is near magnitude -13, and the sun near mag -26.

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Star Catalogs

MOST LUMINOUS STARS IN THE HIPPARCOS CATALOGUE

HIP HD ra (deg) dec (deg) Vmag

Mv =V + 5log(paral

lax in mas) -10

parallaxin mas

std error on

parallax

relative precision

of distance

total proper

motion (mas/yr)

pm in ra(mas/yr

)

pm in dec(mas/

yr)

transverse velocity

in CNS3 Name

24436 34085 78.634 -08.202 0.18 -6.69 4.22 0.81 0.192 1.95 1.87 -0.56 2.19 beta_Ori

100453 194093 305.557 +40.257 2.23 -6.12 2.14 0.51 0.238 2.60 2.43 -0.93 5.76gamma_C

yg

39429 66811 120.896 -40.003 2.21 -5.95 2.33 0.51 0.219 35.09 -30.82 16.77 71.39 zeta_Pup

48002 85123 146.776 -65.072 2.92 -5.56 2.01 0.40 0.199 12.57 -11.55 4.97 29.65upsilon_C

ar

30438 45348 95.988 -52.696 -0.62 -5.53 10.43 0.53 0.051 30.98 19.99 23.67 14.08 alpha_Car

68702 122451 210.956 -60.373 0.61 -5.42 6.21 0.56 0.090 42.21 -33.96 -25.06 32.22 beta_Cen

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Star Catalogs• Both for lost-in-space star identification and for averaging of

random errors, it is desirable to track at least four stars, and preferably more.

• This drives the size of the star catalog.

• The star availability requirement is usually stated as the probability that at least N stars will be available in the tracker’s FOV.

• The number of stars in the FOV can be assumed to follow a Poisson distribution, which says that the probability of finding N stars in the FOV is given by:– λ is the average number of events per interval

– k number of events

– e is the number 2.71828

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Example 4:

!)(

k

ekP

k

• These values can immediately be used to estimate the required size of the star catalog.

• The size of the celestial sphere is 4π(180/π)2= 41253 deg2 . • Example 5:• The required size of the catalog can then be used to estimate the

magnitude range of stars that must be tracked. • Haworth has counted the number of stars in the Tycho star catalog

in visual magnitude ranges from -0.5 to 11.5. • His values for the number of stars of magnitude less than MV for

condition 3.5<=MV<=10.5 can be fitted to within 3% by the simple relation:

– N(MV)=3.9 exp(1.258MV-0.011MV2)

• According to above Equation, a catalog containing 7542 stars must include stars as dim as MV=6.4, while a catalog with 696 stars need only extend to MV=4.3, a magnitude signifying 100.4(6.4-4.3)=~7 times as much energy flux as MV=6.4.

• Example 6:

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Star Catalogs

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Common Coordinate Systems used in Astronomy

o Horizono Equatorialo Ecliptico Galactic

Horizon Coordinate System

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Altitude (a): is the angular distance North (+) or South (-) of the horizon.Azimuth (A) : of a body is its angular distance measured eastwards along the horizon from the north point to the intersection of the object’svertical circle. www.davidreneke.com

Equatorial Coordinate Systems

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In the equatorial system, the Earth's equator and poles are projected outward into space to form a celestial sphere with the Earth at the center. The projection of the Earth's equator onto this sphere is called the celestial equator. Positions of astronomical objects are described by their right ascension or RA (the angle eastward from the vernal equinox) and declination or DEC (the angle above or below the celestial equator). Right ascension is measured in hours, minutes and seconds rather than degrees (as we do with the Earth's longitude). One complete circle is 24 hours rather than 360 degrees. Declination is measured in the same way we measure the Earth's latitude and ranges between -90 and +90 degrees measured from the celestial equator. The equatorial system is the one most commonly used in astronomy.

Angular Distance between two Stars

• 𝑆𝑡𝑎𝑟1 = 𝛼1, 𝛿1• 𝑆𝑡𝑎𝑟2 = 𝛼2, 𝛿2• 𝑐𝑜𝑠 𝛾 = 𝑐𝑜𝑠 900− 𝛿1 𝑐𝑜𝑠 900− 𝛿2 +𝑠𝑖𝑛 900− 𝛿1 𝑠𝑖𝑛 900− 𝛿2 𝑐𝑜𝑠 𝛼1 − 𝛼2

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Example 7:

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In the ecliptic system, latitude is measured with respect to the ecliptic plane and position is measured by ecliptic latitude (elat) and ecliptic longitude (elon). Both ecliptic latitude and longitude are measured in degrees. This system uses the plane of the Earth's orbit around the Sun as a base. Ecliptic coordinates are useful for describing the position of objects within our solar system. Most other objects in space are so far away that their positions in the sky do not change much on a human time scale. This is not true for close objects, like the planets, moons, the Sun, asteroids and comets. So ecliptic coordinates can be more useful than the equatorial system for solar system objects.

Coordinate Systems

https://www.quora.com/

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Galactic Coordinate SystemsThe galactic system uses galactic latitude (glat) and galactic longitude (glon) to define the position of objects in space relative to the galactic center (the center of our galaxy - the Milky Way). Galactic coordinates are measured in degrees. The galactic coordinate system is used to see how astronomical objects are distributed with respect to the galactic plane.

• Equatorial Coordinates

– Right ascension 𝛼

– Declination 𝛿

• Galactic Coordinates

– Longitude (𝑙)

– Latitude (𝑏)

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Transforming between Galactic to Equatorial Coordinates

cos 𝑏 𝑐𝑜𝑠 𝑙 − 330 = cos 𝛿 𝑐𝑜𝑠 𝛼 − 282.250

cos 𝑏 𝑠𝑖𝑛 𝑙 − 330 = sin 𝛿 𝑠𝑖𝑛 62.60 + cos 𝛿 𝑐𝑜𝑠 𝛼 − 282.250 𝑐𝑜𝑠 62.60

sin 𝑏 = sin 𝛿 𝑐𝑜𝑠 62.60 − cos 𝛿 𝑠𝑖𝑛 𝛼 − 282.250 𝑠𝑖𝑛 62.60

cos 𝛿 𝑐𝑜𝑠 𝛼 − 282.250 = cos 𝑏 𝑐𝑜𝑠 𝑙 − 330cos 𝛿 𝑠𝑖𝑛 𝛼 − 282.250 = cos 𝑏 𝑠𝑖𝑛 𝑙 − 330 𝑐𝑜𝑠 62.60 − sin 𝑏 𝑠𝑖𝑛 62.60

sin 𝛿 = cos 𝑏 𝑠𝑖𝑛 𝑙 − 330 𝑠𝑖𝑛 62.60 + sin 𝑏 𝑐𝑜𝑠 62.60

Example 8:

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