Star Coordinates and the Celestial Dome Astronomers have mapped out the sky in the shape of a spherical dome the Celestial Sphere, where earth is just a tiny spec at the center of the dome. The celestial sphere is divided into declination and right ascension just like earth s latitude and longitude lines. Stars are identified by the coordinates of Declination and Right Ascension and are assumed to be fixed as their position don t change on day to day basis. Table 1 provides a list of major constellations and star coordinates. For every Declination point in the sky there is a corresponding Latitude on earth but Longitude and Right Ascension coincide only for a moment each day and drift away as earth spins about its axis. This also means that while the earth equator and celestial equator are on the same plane, the prime meridian and 0 o Right Ascension planes are the same only once a day. Page 1 Where is 0 o Right Ascension? Since the earth is constantly spinning, astronomers had to choose a fixed reference point in time where the 0 o Right Ascension line coincides with the 0 o Longitudinal line.
The time they chose was the Vernal Equinox of March 20, 2000, at 7:30 AM Greenwich Mean Time (GMT). Right Ascension Offset A single rotation of earth about its axis is a little short of 24 hours. It is 23 hours, 56 minutes and 4 seconds or 86164 seconds (also called sidereal time). You can easily calculate Right Ascension offset from any reference longitudinal line, by computing the total number of turns earth has rotated since the 2000 Vernal Equinox, ignore the complete turns but add the remainder degrees to your current location longitude. This value added to the longitude is the reference point Right Ascension. The following is an example calculation for Baton Rouge, LA (30 N, 91 W) Current Time: 09/29/2017 2:30 AM. Vernal Equinox Time: 03/20/2000 7:30 AM. 09/29/2017 2:30 AM - 03/20/2000 7:30 AM = 553114800 seconds. Page 2 Get time difference in seconds from the following URL: (http://starcoordinates.sahraid.com) Earth revolution: 553114800 / 86164 = 6419 turns plus 117 degrees. 117 + (-91) = 26. The reference longitude for Baton Rouge, LA is 26 o. If you were to draw an arrow from the center of the earth to a reference latitude and longitude then extending it into the sky is the respective Declination and Right Ascension angle.
Stars are arc length away from the point of observation as length and direction of arc can be computed geometrically. Great Circle Arc Think of the point of observation P (latitude and longitude) and the star coordinates Q (Declination and Right Ascension) as two points on the celestial sphere. Page 3 The arc length or the angle between the two points is given by the spherical law of cosines. = cos 1 (cos ( o ). cos ( 1 ). cos (λ) + sin ( 1 ). sin ( o ) North Pole as reference direction A reference point R can be added (equal in length of the arc) in the direction of north by adding an arc length to the latitude of the point of observation P and the angle is obtained using the dot product rule as explained in the Workshop section. 3D Compass as a pointing device 3DCompass is a tool to point to the direction of a star in the 3-Dimensional space. The Compass is made of two semi-circle protractors, one for pointing to the Declination and the other for the Right Ascension. The Declination half has a magnetic needle pointing to North and the Right Ascension has a sliding Arrow pointer providing direction in the Sky where star is located. Page 4
As mentioned earlier only the Right Ascension and the Declination angles relative to your reference location is sufficient to locate a star. Just like latitude and longitude are sufficient to point to any place on earth, similarly, Right Ascension and Declination point to any place in the Celestial Hemisphere. The two coordinate systems can be made to coincide with each other by taking in to consideration the degrees of shift in the Right Ascension by earth rotation. Page 5 How to use 3D Compass Obtain the 3DCompass Star Coordinates for the time and location from StarCorrdinates.SahraID.com Fold the compass so the Right Ascension is vertical to the declination half. Hold the Compass so the Declination 0 o is horizontal facing north. Position the Right Ascension pointer at the mark of Right Ascension Arc. Rotate the compass horizontally until the Magnetic-needle North points to the Declination Arc. The pointer points to the star. Page 6
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Workshop Exercise 1 Calculate the number of revolutions earth has made since Vernal Equinox of Mar 20, 2000 7:30 AM (GMT) till September, 29, 2:30 AM. (earth takes 23 hours, 56 minutes and 4 second to revolve around its own axis.) 9/20/2017 2:30 AM 3/20/2000 7:30 = 553622400 seconds. Earth revolution: 553114800 / 86164 = 6419 20 turns plus 117 degrees. Exercise 2 Calculate the Arc length of star Betelgeuse from earth latitude 30 o, longitude -91 o at the time October, 4, 11:30 PM. (The celestial coordinates of star Betelgeuse are 8 o Declination, 88.5 o Right Ascension.) The Arc indicates the Right Ascension angle for 3D Compass. Earth has shifted 117 o from Celestial coordinates (see Exercise 1). Reference longitude = -91 + 117 = 26 o, Reference latitude = 30 o. Point P (30,26). Page 8 Star Declination = 8 o, Star Right Ascension 88.5 o. Point Q (8, 88.5). Δφ = cos -1 (cos(φ1) x cos(φ2) x cos(λ) + sin (φ1) x sin(φ2)) Δφ = cos -1 (cos(30) x cos(8) x cos(88.5-26)) + sin (30) x sin(8))
Δφ = 62.5 o Exercise 3 Draw a point facing north equal distance of arc length from reference location (30,26) to star coordinate (8,88,5). Add the arc length 62.5 o (from exercise 2) to the reference latitude. Point R = ((30 + 62.5), 26) = 92.5 o,26 o Exercise 4 Convert polar coordinates of point P, R and Q into rectangular coordinates Point P (26,30), Page 9 Px = cos(30)x(cos(26) = 0.78, Py=cos(30)xSin(30) = 0.38, Pz = Sin(30) = 0.5. Point Q (8,88.5), Qx = cos(8)x(cos(88.5) = 0.026, Qy=cos(8)xSin(88.5) = 0.99, Qz = Sin(8) = 0.14. Point R (92.5,30),
Rx = cos(8)x(cos(88.5) = -0.04, Ry=cos(8)xSin(88.5) = -0.017, Rz = Sin(8) = 0.99 Exercise 5 Convert to Vector PR (point P and point R) and convert to Vector PQ (Point P and point Q) Vector PR X = R x P x Y= R y- P y, Z = R z - P z X = -0.04 0.78 = -0.81, Y = -0.017 0.38 = -0.4 Z = 0.5 Vector PQ X = Q x P x Y= Q y- P y, Z = Q z - P z X = 0.026 0.78 = -0.75, Y = 0.99 0.38 = 0.61 Z = -0.36 Page 10 Exercise 6 Compute Vector magnitude PR and PQ. Magnitude of Vector PR = Magnitude of Vector PQ = x 2 + y 2 + z 2 = 0. 81 2 + 0. 4 2 + 0. 5 2 = 1. 03 x 2 + y 2 + z 2 = 0. 75 2 + 0. 61 2 + 0. 36 2 = 1. 03
Exercise 7 Compute dot product of the Vector PR and PQ. DotProduct = PR PQ = (PRx PQx + PRy PQy + PRz PQz) = PR PQ = ( 0.81 0.75 + 0.4 0.61 + 0.5 0.36) = 0.19 Exercise 8 Compute angle between the Vector PR and PQ. Page 11 α = cos 1 dot product magnitude A Magnitude B α = cos 1 0.19 = 80 degree 1.03 1.03 Exercise 9 Compute angle between the vector PQ and the plane normal to the vector PR. (This is the declination angle for 3D Compass).
= 90 - Trigonometric Ratios Trigonometry is the study of triangles, their measurements, and calculations based on these measurements. The right triangle has a special place in trigonometry because of its interesting characteristics. The Pythagoras principal states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of its sides. This leads to some special ratios like sine, cosine and tangent, which are used extensively in such fields as science, engineering, and architecture. Trig Functions Sine: In a right triangle, the ratio of the length of the opposite side to its hypotenuse (for a given angle θ or theta) is a constant and is called sin θ or the sine of the angle θ. Its inverse trig function is cosecant, denoted csc θ. sin θ = Opposite Hypotenuse csc θ = 1 sin θ = Hypotenuse Opposite Page 13 The sine of an angle in a right triangle can also be a measure of steepness. Imagine a road that uniformly rises 6 feet in height for every 100 ft in length. The steepness of the road is then 6. Coincidentally, the measure of 100 steepness is important in road construction; any value greater than 6 is usually avoided for safety reasons. 100
Cosine: In a right triangle, the ratio of the length of its adjacent side or base to its hypotenuse is a constant for any angle θ, and is called cos θ or the cosine of the angle θ. Its inverse trig function is secant, denoted sec θ. cos θ = Adjacent Hypotenuse sec θ = 1 = Hypotenuse cos θ Adjacent Page 14 Tangent: In a right triangle, the ratio of the length of its opposite side to its adjacent side or base is a constant for any angle θ, and is called tan θ or the tangent of the angle θ. Its inverse trig function is cotangent, denoted cot θ. tan θ = Opposite Adjacent cot θ = 1 = Adjacent tan θ Opposite Inverse Trig Functions Sine, cosine, and tangent give the ratios of sides in a right triangle for a given angle. But what if you know the ratio and want to find the measure of the angle? Inverse trigonometric functions solve this problem by giving an angle in a right triangle from a given ratio. Arcsine: Arcsine gives the angle θ given the sine ratio. It is denoted arcsin x or sin 1 x. When the ratio of the opposite side to the hypotenuse is given, the corresponding measure of A is given by the arcsine of the opposite over the hypotenuse.
sin θ = Opposite Hypotenuse θ = sin 1 ( Opposite Hypotenuse ) Arccosine: Arccosine gives the angle θ given the cosine ratio. It is denoted arccos x or cos 1 x. When the ratio of the adjacent side to the hypotenuse is given, the corresponding measure of A is given by the arccosine of the adjacent side to the base. cos θ = Adjacent Hypotenuse θ = cos 1 ( Adjacent Hypotenuse ) Page 15 Arctangent: Arctangent gives the angle θ given the cosine ratio. It is denoted arctan x or tan 1 x. In other words, when the ratio of the opposite side to the adjacent side is given, the corresponding measure of A is given by the arctangent of the opposite side to the adjacent side. tan θ = Opposite Adjacent θ = tan 1 ( Opposite Adjacent ) Complimentary Relationships of Trig Ratios In a right triangle, the two angles opposite to the base and the perpendicular are complimentary. For example, in Fig 2.1, the angles α and β satisfy α + β = 90 = π 2 radians. Fig 2.1: Angles α and β are complementary The nature of the trigonometric functions implies that the following statements are true:
cos α = sin ( π α) = sin β 2 csc α = sec ( π α) = sec β 2 cot α = tan ( π α) = tan β 2 Page 16 The Unit Circle A perpendicular drawn from the perimeter of a circle of radius 1 to the x-axis makes a right triangle, as shown in Fig 2.2. This circle is known as the unit circle. From this triangle, squares of trigonometric ratios can be obtained using the Pythagorean Theorem. Fig 2.2: Squares of the sides of a right triangle formed on the unit circle. With reference to Fig 2.2, we can simplify the trigonometric ratios by substituting 1 (the radius of the circle) as the length of the hypotenuse. sin θ = Opposite = a cos θ = Base = b Using the Pythagorean Theorem, we can obtain the following important identities: sin 2 θ + cos 2 θ = 1 sin 2 θ = 1 cos 2 θ or sin θ = 1 cos 2 θ cos 2 θ = 1 sin 2 θ or cos θ = 1 sin 2 θ Page 17 tan θ = sin θ cos θ
Worksheet: Computing Sine and Cosine Using a Circle While scientific calculators are able to determine sine and cosine automatically, it s also possible to compute the ratio graphically from the triangles formed as shown in Fig 2.3. Note: The hypotenuse in all of the triangles is 40 units, and the into 10 angles. a = opposite, b = adjacent, and c = quarter circle (90 ) is divided hypotenuse. Trig Ruler A trigonometric ruler can be constructed that shows the values of the trigonometric functions without the use of a calculator. The ruler consists of three scales, labeled Hypotenuse, Base, and Perpendicular, and a circular disc measuring angles around a circle from 0 to 360. The length of each scale is 1 unit, but it is also divided into 100 smaller units. Page 18 To assemble the trig ruler, cut out the scales and the circular disc and attach as shown in Fig 2.5. (Note: the perpendicular is suspended from the other end of the hypotenuse.)
Fig 2.4: Trig Ruler scales and circular disc Fig 2.5: Trig Ruler assembly Page 19 Using the Trig Ruler Measuring the Sine of an Angle: Measuring sin 30. First, align the hypotenuse to 30. Then, align the freely suspended Perpendicular to the Base so that the Base and the Perpendicular form a right angle. Read the value on the Perpendicular scale where the Perpendicular and Base scales intersect; this will be the sin of 30. Measuring the Cosine of an Angle: Measuring cos 30. Align the scales as shown in the previous example, but this time read the value from the Base scale where the Perpendicular and Base scales intersect rather than from the Perpendicular scale. Measuring the Tangent of an Angle: Measuring tan 30. Align the scales as shown in the previous examples, but instead divide the value on the Perpendicular scale by the value of the Base scale since tan θ = sin θ cos θ.
Activity: Measuring Height and Distance Using the Trig Ruler You can calculate the height of an object (such as a building or a tree) and its distance with the help of the Trig Ruler by simply measuring the angle of your line of sight. Page 20 To find out how far away an object is, stand at a distance and align the Trig Ruler such that the base is horizontal and parallel to the ground and the hypotenuse is aligned toward the bottom of the object, as shown in Fig 2.6. By doing this, you are creating two similar triangles. Using the fact that the sides of similar triangles are proportional, you can find the object s distance x from you. Note: You are looking at the object from your height, so the height h is same as your height. The height h is the reading from the Trig Ruler s Perpendicular scale and the distance x is the reading from the Trig Ruler s Base scale (where the Base and the Perpendicular intersect), x is the distance from the object. Page 21
This gives you your distance from the object, x h = x h Fig 2.6: Measuring distance using Trig Ruler. The height H of the object can be calculated by aligning the hypotenuse toward the top of the object and measuring the Trig Ruler s Base and Perpendicular scales at the point where they intersect. If the base scale measures x and the perpendicular scale measures h, then the height of the object can be found by using: H h = X x Fig 2.7: Measuring height of an object with Trig Ruler. Page 22
Sky Map Observing the night sky at the same time every month, it appears to shift 30 degrees from month to month. This phenomenon occurs because the Earth completes a single rotation about its axis in 23 hours, 56 minutes and 4 seconds, while it takes a full 24 hours for a fixed location on earth to face the sun again. These 4 extra minutes in a day account for one degree of rotation. Therefore, in 30 days the earth rotates 30 extra degrees. The SkyMap Guide provides a month to month correlation of the celestial map to the earth map. 12 Month Sky Chart We can predict which segment of the night sky will be over our heads given a specific time and location. Since the sky shifts only a single degree each day, these shifts are not visibly significant on a day to day basis. However, there is a significant 30 o change from month to month. Figure 1 through Figure 12 show monthly geographical maps of the earth viewed from space at the same time of night; There is a 30 o frame shift each month. Sky Map Overlay If we were to place an overlay of the sky map over the earth map shown in Figure 13, we would see the earth shift nearly 30 o every month. In order to match the SkyMap Overlay to the earth map you need to know the month and time of day. Think of Figure 1 as a snapshot of the earth taken from the Sky at 0 o Right Ascension and 0 o Declination. How to use Sky Map Overlay Suppose it is Aug 2, 2017 10:34 PM and our current location s coordinates are latitude 0 and longitude 0. The SkyMap Overlay, when placed on the earth map of Figure 1 (August), will match at every point. At the same time and location thirty days later, the earth map of Figure 2 (September) will match point to point with the overlay. For the month following, Figure 3 will match the overlay at each point and so on and so forth. This pattern repeats every year. If your local time is two hours ahead of GMT, refer back to the previous month s figure. If your local time is two hours behind GMT, refer to the following month s figure and the Sky Map Overlay will match your location. Table 2 describes mapping of the figures with time zones.
Table 2. Mapping of the overlay with respect to Time Zone Januray February March April May June July August September October November December GMT(+11Hours) Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 GMT(+10Hours) Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 GMT(+9Hours) Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 GMT(+8Hours) Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 GMT(+7Hours) Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 GMT(+6Hours) Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 GMT(+5Hours) Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 GMT(+4Hours) Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 GMT(+3Hours) Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 GMT(+2Hours) Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 GMT(+1Hours) Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 GMT(+0Hours) Figure6 Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 GMT(-1Hours) Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 GMT(-2Hours) Figure7 Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 GMT(-3Hours) Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 GMT(-4Hours) Figure8 Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 GMT(-5Hours) Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 GMT(-6Hours) Figure9 Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 GMT(-7Hours) Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 GMT(-8Hours) Figure10 Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 GMT(-9Hours) Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10 GMT(-10Hours) Figure11 Figure12 Figure1 Figure2 Figure3 Figure4 Figure5 Figure6 Figure7 Figure8 Figure9 Figure10