Introduction in Computational Astronomy Building a Space Simulator

Transcription

1 Introduction in Computational Astronomy Building a Space Simulator Marc Eduard Frincu - mfrincu@info.uvt.ro West University of Timisoara

2 Outline 1 Introduction

3 Outline 1 Introduction 2 General theoretical notions Prequisites Points, lines, circles and spheres Coordinate systems Main angles Time Conversions between coordinate systems

4 Outline 1 Introduction 2 General theoretical notions Prequisites Points, lines, circles and spheres Coordinate systems Main angles Time Conversions between coordinate systems 3 Planetary and stellar coordinates Stellar coordinates Planetary coordinates Planetary ephemerisis Comets and asteroids position Computing the rise-set times

5 Outline 1 Introduction 2 General theoretical notions Prequisites Points, lines, circles and spheres Coordinate systems Main angles Time Conversions between coordinate systems 3 Planetary and stellar coordinates Stellar coordinates Planetary coordinates Planetary ephemerisis Comets and asteroids position Computing the rise-set times 4 Starcharts Building a stereographic polar projection map

6 Outline 1 Introduction 2 General theoretical notions Prequisites Points, lines, circles and spheres Coordinate systems Main angles Time Conversions between coordinate systems 3 Planetary and stellar coordinates Stellar coordinates Planetary coordinates Planetary ephemerisis Comets and asteroids position Computing the rise-set times 4 Starcharts Building a stereographic polar projection map 5 Copernicus

7 Introduction Definition Computational Astronomy : astronomy done using a computer

8 Introduction Definition Computational Astronomy : astronomy done using a computer Applicability

9 Introduction Definition Computational Astronomy : astronomy done using a computer Applicability modern astronomy relies on huge amounts of data from large sky surveys which needs to be processed; task impossible for humans need for the development and application of methods for archiving, analysing and modelling the data

10 Introduction Definition Computational Astronomy : astronomy done using a computer Applicability modern astronomy relies on huge amounts of data from large sky surveys which needs to be processed; task impossible for humans need for the development and application of methods for archiving, analysing and modelling the data increase need to represent the skies and model the universe need for fast computers to deliver real time views

11 Introduction Definition Computational Astronomy : astronomy done using a computer Applicability modern astronomy relies on huge amounts of data from large sky surveys which needs to be processed; task impossible for humans need for the development and application of methods for archiving, analysing and modelling the data increase need to represent the skies and model the universe need for fast computers to deliver real time views educational purposes such as: presentations, atlases, e-books, etc.

12 Introduction Definition Computational Astronomy : astronomy done using a computer Applicability modern astronomy relies on huge amounts of data from large sky surveys which needs to be processed; task impossible for humans need for the development and application of methods for archiving, analysing and modelling the data increase need to represent the skies and model the universe need for fast computers to deliver real time views educational purposes such as: presentations, atlases, e-books, etc....

13 Introduction Concrete examples

14 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc.

15 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc. cosmology: mapping of dark matter in the Universe directly through gravitational lensing surveys, etc.

16 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc. cosmology: mapping of dark matter in the Universe directly through gravitational lensing surveys, etc. celestial mechanics: N-body problem, planetary ephemerisis, stellar coordinates etc.

17 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc. cosmology: mapping of dark matter in the Universe directly through gravitational lensing surveys, etc. celestial mechanics: N-body problem, planetary ephemerisis, stellar coordinates etc. space simulators: Celestia, OpenUniverse, Copernicus, etc.

18 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc. cosmology: mapping of dark matter in the Universe directly through gravitational lensing surveys, etc. celestial mechanics: N-body problem, planetary ephemerisis, stellar coordinates etc. space simulators: Celestia, OpenUniverse, Copernicus, etc. virtual planetariums: StarCalc, etc.

19 Introduction Concrete examples astrophysics: precision spectra of galaxies based on population synthesis modelling, etc. cosmology: mapping of dark matter in the Universe directly through gravitational lensing surveys, etc. celestial mechanics: N-body problem, planetary ephemerisis, stellar coordinates etc. space simulators: Celestia, OpenUniverse, Copernicus, etc. virtual planetariums: StarCalc, etc....

20 Theoretical notions Why? Because without them we would not understand astronomy. What will we know after this section?

21 Theoretical notions Why? Because without them we would not understand astronomy. What will we know after this section? the most important points, lines, circles and distances used in astronomy

22 Theoretical notions Why? Because without them we would not understand astronomy. What will we know after this section? the most important points, lines, circles and distances used in astronomy the main angles used in astronomy and how to compute them

23 Theoretical notions Why? Because without them we would not understand astronomy. What will we know after this section? the most important points, lines, circles and distances used in astronomy the main angles used in astronomy and how to compute them a litle bit more about time

24 Prequisites What do we need to know before proceeding? Basic spherical geometry spherical triangle (sum of angles greater than 180 degrees) Sine laws Cosine laws Napier s relations Borda s formula Delambre s formula

25 Points, lines, circles and spheres used in astronomy Definition The local vertical direction pointing away from direction of the force of gravity at that location, i.e. the direction pointing directly above a particular location is called zenith. Fact Zenith is also used for the highest point reached by a celestial body during its apparent orbit around a given point of observation.

26 Points, lines, circles and spheres used in astronomy Definition The local vertical direction pointing away from direction of the force of gravity at that location, i.e. the direction pointing directly above a particular location is called zenith. Fact Zenith is also used for the highest point reached by a celestial body during its apparent orbit around a given point of observation. Definition The vertical direction at the given location and pointing in the same sense as the gravitational force, i.e. the direction pointing directly under a particular location is called nadir.

27 Points, lines, circles and spheres used in astronomy

28 Points, lines, circles and spheres used in astronomy Definition The Earth s axis of rotation is an imaginary line that goes through the center of the sphere which makes up the Earth and around which the Earth itself rotates.

29 Points, lines, circles and spheres used in astronomy Definition The Earth s axis of rotation is an imaginary line that goes through the center of the sphere which makes up the Earth and around which the Earth itself rotates. Definition The celestial sphere is represented by the imaginary sphere with infinite radius on which all the cosmic objects seem to be located.

30 Points, lines, circles and spheres used in astronomy Definition The celestial poles are the two imaginary points in the sky where the Earth s axis of rotation, infinitely extended, intersects the celestial sphere. Fact The celestial poles are NOT the same as the Earth s magnetic poles!

31 Points, lines, circles and spheres used in astronomy Definition A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two equal hemispheres.

32 Points, lines, circles and spheres used in astronomy Examples of great circles On Earth (can you think of any?):

33 Points, lines, circles and spheres used in astronomy Examples of great circles On Earth (can you think of any?): the equator : great circle having the axis of rotation perpendicular on it divides the Earth into two equal hemispheres: north and south the meridians : great circles passing through the Earth s rotation axis; a special meridian is called local meridian it is the meridian passing through the observer s position. You can quickly determine it by using a telescope as follows: point it to the North Star (!) and fix its position so that it cannot be moved horizontaly (leave the vertical movement accesible). The imaginary line (circle) connecting the telescope with the North Star is called local meridian. hour circle : great circle passing through the celestial poles and the position of the observer

34 Points, lines, circles and spheres used in astronomy More examples of great circles On the celestial sphere (any guess?):

35 Points, lines, circles and spheres used in astronomy More examples of great circles On the celestial sphere (any guess?): the astronomical horizon - the plane passing through the observer s eye, parallel with the plane tangent to Earth in the position of the observer (geodesic horizon) [none of the listed horizons is the true horizon i.e. the place where the Earth s surface meets the celestial sphere] the celestial equator - great circle on the imaginary celestial sphere, in the same plane as the Earth s equator the ecliptic - the apparent path that the Sun traces out in the sky during the year

36 Different horizons

37 Points, lines, circles and spheres used in astronomy Definition A small circle of a sphere is the circle constructed by a plane crossing the sphere not in its center. Examples the latitude circles

38 Points, lines, circles and spheres used in astronomy

39 Coordinate systems Definition A coordinate system is a way of determining the position of a body Definition The vernal equinox point is the point where the ecliptic intersects the equatorial plane in the moment when the Sun is crossing from the Southern celestial hemisphere to the northern one.

40 Coordinate systems Examples of coordinate systems Hour coordinates Spherical coordinates Horizontal coordinates Ecliptical coordinates Heliocentrical coordinates Galactic coordinates

41 Coordinate systems Fact From now on we shell asume to use an equatorial coordinate system where we have the following characteristics: center : Earth s center poles : celestial poles distance angle : declination (?) direction angle : right ascension (?) marker point : vernal equinox main circle : celestial equator

42 Coordinate systems Fact Another important coordinate system that we will use is the horizontal coordinate system: center : point of observation poles : zenith and nadir distance angle : altitude (?) direction angle : azimuth (?) marker point : north point of the horizon main circle : horizon

43 Main angles Some important angles the hour angle - HA measured in degrees the right ascension - α or RA measured in hours the declination - δ or Dec measured in degrees the Greenwitch hour angle - GHA and in particular the GHA Aries which is the GHA of the vernal equinox both measured in degrees Greenwitch sidereal time GST is same as GHA Aries but measured in hours the Sidereal hour angle - SHA measured in degrees

44 Main angles

45 Time Fact There are lots of ways to compute time and many different times in astronomy : Universal Time (UT) or GMT, Universal Time Coordinated (UTC), Terestrial Dynamic Time (TDT), Legal Time (LT), Local Sidereal Time (LST), etc. Fact Also there is more than one notion for day or year: Sidereal Day, Real Solar Day, Mean Solar Day, Sidereal Year, Tropical Year, etc.

46 Time Fact There are lots of ways to compute time and many different times in astronomy : Universal Time (UT) or GMT, Universal Time Coordinated (UTC), Terestrial Dynamic Time (TDT), Legal Time (LT), Local Sidereal Time (LST), etc. Fact Also there is more than one notion for day or year: Sidereal Day, Real Solar Day, Mean Solar Day, Sidereal Year, Tropical Year, etc. Local Sidereal Time We are interested in LST because it plays an important part in computing planetary coordinates and when converting from one coordinate system to another

47 LST

48 Conversions between coordinate systems From the equatorial system to the horizontal system We need to know: the time, the date, te location of the observer (latitude and longitude) and the equatorial coordinates of the celestial body (α and δ). All the above data needs to be expressed in degrees except for the time which needs to be expressed in hours. For finding out the altitude and azimuth of the body we need first to compute the LST and the HA as follows:

49 Conversions between coordinate systems Computing the LST LST = d + long + 15 UT where d is the day number (i.e. the no of days since Jan 1st :00:00), long is the longitude of the observer and UT is the Universal Time IMPORTANT LST [0..360] degrees To get the LST expressed in hours divide by Computing the HA HA = LST α IMPORTANT HA [0..360] degrees

50 Conversions between coordinate systems Finding the Altitude - Alt sin(alt) = sin(δ) sin(lat) + cos(δ) cos(lat) cos(ha) from which we obtain directly alt. Finding the Azimuth - Az cos(a) = sin(δ) sin(alt) sin(lat) cos(alt) cos(lat) In the case sin(ha) < 0 we have Az = A, otherwise we get Az = 360 A

51 Planetary and stellar coordinates Why do we need them? Because astronomers need planetary and stellar coordinates to extrapolate and build models with the purpose to enhance the knowledge about the motion of cosmic objects we can deduce how the solar system s or stars configuration looked eons ago or will look in the near/far future we can predict collisions between bodies (Shoemaker-Levi comet 96)

52 Stellar coordinates They help us understand and view the cosmic objects (i.e. starts, galaxies, nebula, clusters, etc.) as they really are located in space How is this achieved? By converting the spherical coordinates (the angles α, δ and distance to the object ρ) into orthogonal coordinates (x,y,z). Note: the reference point can be either the Sun or the galactic center as we will see in the following slides.

53 Revisiting the RA and Dec Declination Is the latitude on the celestial sphere of the body in question Fact One can obtain it by using a telescope parallel with the horizontal as follows : observe the star when it passes through the local meridian and measure the angle with regard to the horizontal. The angle will give the object s δ

54 Revisiting the RA and Dec Right Ascension Is represented by the angular distance between meridian 0 and the meridian of the observed star (in celestial coordinates). Fact To determine it is suffices to know the α of at least one star. Simply observe the moment when the star with the known α passes through the local meridian and write down the exac time. Then observe the time when the star you are interested in passes through the local meridian and also mark it down. The difference between the two times will give you the difference between the two s αs.

55 Finding the orthogonal coordinates Steps convert α and δ in degrees: φ = α hours 15 + α mins α secs θ = δ degrees + ( δ mins 60 + δsecs 3600 ) sign(δ degs)

56 Finding the orthogonal coordinates Steps convert α and δ in degrees: φ = α hours 15 + α mins α secs θ = δ degrees + ( δ mins 60 + δsecs 3600 ) sign(δ degs) compute the (x,y,z) coordinates: rvect = ρ cos(θ) x = rvect cos(φ) y = rvect sin(φ) z = ρ sin(θ)

57 And in Galactic Coordinates Fact These coordinates are not galactic coordinates. To obtain those one needs to use the following formula (GLIESE ): x g = ( x) ( y) ( z) y g = ( x) ( y) + ( z) z g = ( x) ( y) + ( z) where the galactic center is located at α g = 17h45.6m, δ g = 28d56.3m, and the galactic north pole at α g = 12h51.4m, δ g = 27d07.7m

58 What else can I do with them? Besides a nice 3D view one can achieve much more:

59 What else can I do with them? Besides a nice 3D view one can achieve much more: rotations, translations, scaling (for better viewing) convert back from orthogonal to equatorial coordinates find out the magnitude of a star as seen from another place in the galaxy find out what the equatorial coordinates of the stars would look like if you were to be located near another star find out how the stars configuration would look like in the future or it has looked like in the past (needs the radial speed, and the proper motion)

60 Planetary coordinates To compute the positions of the bodies (planets, satellites, comets, asteroids, etc.) in the Solar System one needs to know their orbital elements. There are six main orbital elements (known from astronomical tables, books, atlases): longitude of the ascending node (N) - the angle across the ecliptic from the vernal point to the ascending node (i.e. the intersection between the body s orbit and the ecliptic at the time the body leaves the southern part of the ecliptic and enters the northen one) the inclination (i) - the angle of the orbit with regard to the ecliptic argument at perihelion (ω) - angle between the ascending node and perihelion

61 Planetary coordinates major semiaxis (a) - the major semiaxis of the orbit excentricity (ǫ) mean anomaly (M) - 0 at the perihelion and 180 degrees at the aphelion. Can be computed if we know the orbital period (P) and the time since perihelion (T): M = n (t T) = (t T) 360 P

62 View on the main orbital elements

63 Planetary coordinates Other elements can also be used or derived from the main six: longitude of perihelion : ω 1 = N + ω mean longitude : L = M + ω 1 distance at perihelion : q = a (1 ǫ) orbital period : P = a m, where m is the body s mass (0 for comets and asteroids) time at perihelion : T = Epoch M 360 P true anomaly (v) : angle between periheliona dn the body s position excentric anomaly (E) angle introduced by Kepler in it eqs in order to compute v based on M and ǫ daily motion : 360 P

64 Generalization of Kepler s laws and the Newton s Universal Attraction Law A little bit of history in 1619 J. Kepler discovered the three famous laws governing the motion of bodies around the Sun sir I. Newton ( ) has extended these laws by assuming that there is a law of attraction between every two bodies in the Universe and that force has the form: F = K m 1 m 2 r 2 12

65 View on Kepler s laws Kepler s laws The orbit of every planet is an ellipse with the sun at one of the foci A line joining a planet and the sun sweeps out equal areas during equal intervals of time The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits

66 The true anomaly v Kepler s equations E ǫ sin(e) = M tan( v 2 ) = 1+ǫ 1 ǫ tan( E 2 ) Deducing v from Kepler s eqs is very important as it is related to the Kepler s 2nd law and esential when trying to figure out the position of a solar system body around the Sun. It can be computed in several ways but the easiest is a simple iterative numerical method. Despite its simplicity this method is also responsible for introducing errors.

67 The true anomaly v Iterative method for finding v Start with: E 0 = M Continue iteratively until the diffence between two consecutive Es is smaller than a given value e: E 1 = M + ǫ sin(e 0 ) E 2 = M + ǫ sin(e 1 )... The last E gives you the value needed

68 Position of the Sun Fact The true position when computing the Sun s coordinates are the one of Earth on orbit around the Sun. Because we view everything from a geocentric perspective we can imagine that in fact the Sun is moving around Earth. Steps in computing the Sun s position Step 1 : compute E from M and ǫ Step 2 : compute the distance r from Earth and v Step 3 : compute the true longitude and convert it and r into rectangular geocentric coordinates Step 4 : convert the previously obtained coordinates into equatorial coordinates Step 5 : compute α and δ of the Sun

69 Position of the Sun And the math... Step 1 : E = M + ǫ sin(m) (1.0 + e cos(m)) Step 2 : x v = r cos(v) = cos(e) ǫ y v = r sin(v) = 1.0 ǫ 2 sin(e) v = tan2 1 (x v,y v ) r = r s = xv 2 + yv 2 Step 3 : long Sun = v + ω x s = r cos(long Sun ) y s = r sin(long Sun )

70 Position of the Sun And the math... Step 4 : x e = x s y e = y s con(ecl) z e = y s sin(ecl) Step 5 : α = tan2 1 (y e,x e ) δ = tan2 1 (z e, x 2 e + y 2 e )

71 Position of the planets and the Moon Fact For the Moon the position is actually the geocentric position (from the center of Earth) not the real topocentric one (from the surface of Earth). To compute the topocentric position some aditional adjustments need to be done. In the case of the planets the computed position is the heliocentric one.

72 Position of the planets and the Moon Steps in computing the planets and Moon s position Step 1 : compute E from M and ǫ Step 2 : compute the distance r from Earth and v Step 3 : compute the position in space (rectangular heliocentric - planets, and geocentric - Moon, coordinates) ates Step 4 : readjust coordinates so that perturbations from the major planets (Jupiter, Saturn and Uranus) is taken into consideration Step 5 : compute the geocentric coordinates of the planets Step 6 : compute α and δ from the geocentric coordinates Step 7 : (only for the Moon) compute the topocentric position

73 Position of the planets and the Moon And the math... Step 1 : See Sun Step 2 : See Sun Step 3 : x h = r (cos(n) cos(v + ω) sin(n) sin(v + ω) cos(i)) y h = r (sin(n) cos(v + ω) + cos(n) sin(v + ω) cos(i)) z h = r sin(v + ω) sin(i) Step 4 : formulas for perturbations Step 5 : x s = r s cos(long Sun ) y s = r s sin(long Sun ) and finally: x g = x h + x s y g = y h + y s z g = z h

74 Position of the planets and the Moon And the math... Step 6 : x e = x g y e = y g cos(ecl) z g sin(ecl) z e = y g sin(ecl) + z g cos(ecl) from which results by (Step 6 - Sun) α and δ NOTE one can compute the geocentric radius using the above coordinates : r g = x 2 e + y 2 e + z 2 e or from the geocentric coordinates in the manner. Step 7 : formulas for computing the topocentric Moon coordinate

75 Planetary ephemerisis Once one knows the position of an object some other interesting things might become useful: apparent diameter (D) : D = D 0 r, where D 0 is the apparent diameter as seen from 1 a.u. elongation (elong) : gives us the angular distance of the planet from the Sun; when smaller than 20 degrees the planet becomes difficult to observe, and when it goes bellow 10 degrees it is not visible anymore magnitude (m): is how bright the planet is compared to a refference object. It s computation differs from one body to another and depends on the fv phase angle (fv) : tells us the planet s phase; 0 degrees for full phase and 180 degrees for new phase

76 Planetary ephemerisis And the math... For elongation : elong = cos 1 ( r2 s +r 2 g r 2 h 2 r s r g ) For the phase angle : fv = cos 1 ( r2 h +r2 g rs 2 2 r g r h ) from where we can compute the phase : phase = 1+cos(fv) 2 For magnitude : (Example for Saturn) m = log 10 (r h r g ) fv +m rings where m rings = 2.6 sin( B ) sin 2 (B) and B is the rings tilt. IMPORTANT In the case of the Moon the computations are different as they would induce to many errors if dealt with as above.

77 Computing the position in space for asteroids and comets These cases are of particular type because: for asteroids N is missing as their orbit constantly changes; M can also be given for a different day and as a result needs to be updated by using P N = N old ( Epoch) d M = M P for comets M is not given as they have eliptical orbits, instead we have T which we can use to compute M (M = 360 d dt P ); another element that can miss is a, but it can be computed from q Note a particular case is that of those having parabolic orbits. For them P =, M = 0, a =. In this case a different approach needs to be taken.

78 Computing the rise-set times There are three possible cases for a celestial body: Steps circumpolar (never sets) : lat body δ body > 0 and lat body + δ body > 90 rises and sets never rises : lat body δ body < 0 and lat body δ body > 90 compute LHA : cos(lha) = sin(h) sin(lat body ) sin(dec body ) cos(lat b ody) cos(dec body ) where h = π 180 radians compute UT sun in south = α GMSTO long body 15.0 where GMSTO = L and L = M + ω

79 Computing the rise-set times Interpretation in case the arccos(lha) < 1 the body is circumpolar in case the arccos(lha) > 1 the body never rises otherwise time rise = UT sun in south LHA (in hours) and time set = UT sun in south + LHA (in hours)

80 Building a planisphere When one builds a starchart it actually creates a 2D representation of the celestial sphere. There are many ways of projecting but the easiest is the stereographic projection: polar or equatorial. There are two main advantages on why to use it: all the circles on the sphere are drawn as circles on the projection plane the angles and small bodies are being conserved the contellations figures remain familiar near the horizon; only bad thing is that they get bigger Of course planispheres are not the only thing one can draw. For example with an equatorial stereographic projection one can create slices of the celestial sphere

81 Building a starchart What do one needs to draw a planisphere? firstly the polar stereographic projection must be chosen as means of representation compute the x and y coordinates from α and δ: x = cos(az) tan( 90 alt 2 ) x = sin(az) tan( 90 alt 2 ) where az and alt are the altitude and azimuth of the object and a method for computing them from α and δ has been presented earlier given a great circle of constant α or a small circle of constant δ compute the radius and center of the circles which correspond to the projection plane given an arc of certain length on either a great or small circle compute its length on the projection plane

82 Copernicus Space Simulator v1.6 Developed by Marc Frincu : present

83 Copernicus About 2004 : started as a personal project (C and OpenGL) 2005 : presented at the Romanian Academy Days 2006 : provided the basis for my thesis 2008 : ported to C# and DirectX 9 plus more features Features moving through the Solar System and the Local Space Constellation lines and Deep Sky Objects image billboards in Solar System view 3D view of planetary surfaces change the time Rightarrow view Solar System configuration in past or future realistic comet tails and 3D meshes for comets and asteroids

84 Copernicus Features (continued) lens flare for Sun light, Deep Sky billboards GoTo functions for Solar System objects possibility to define your own custom obects 2D view of the solar system polar projection map Fact The application allows the loading of 87,475 stars, 168 Deep Sky Objects, 500 trans-neptunes, 200 asteroids, 400 comets and 30 planetary moons at any given moment. Despite these restrictions, the actual databases that come with the program include 10,986 asteroids and 755 trans-neptunes - for memory issues.

85 Example : 3D view of a comet

86 Example : 2D view of the Inner SS

87 Example : Polar Projection Map

88 Example : 3D view of a planet s surface (orbit)

89 Example : 3D view of a planet s surface (near view)

90 Next? more features? case studies? (computational paralelism, improved mathematical methods, etc.) VR?...

91 Thank you Questions? Remarks?