Astronomical Coordinates, Distances and Magnitudes
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Volume 1 INTRODUCTION TO ASTRONOMY Astronomical Coordinates, Distances and Magnitudes Author: Prof. Ana Inés Gómez de Castro Universidad Complutense de Madrid Astronomy Workshop i INTRODUCTION TO ASTRONOMY Astronomy Workshop © Ana Inés Gómez de Castro Facultad de Ciencias Matemáticas Universidad Complutense de Madrid email: [email protected] Table o f Contents CHAPTER 1 Reference Systems (RSs) 1 Classification of Astronomic RSs 3 Fundamental planes 3 Alt-Azimutal System 5 Absolute Equatorial System 6 Hour Equatorial System 7 Ecliptic System 7 Summary 8 CHAPTER 2 Distances and Magnitudes 9 Units of Distance 9 Astronomic Magnitudes 10 Absolute Magnitudes 13 Summary 14 CHAPTER 3 Radiation Pressure 15 Definition 15 Spacial Missions 15 Summary 17 APPENDIX 1 Table of Exoplanets 18 Chapter 1 Reference Systems Cartesian and Spherical Polar Systems of Reference n order to locate the position of a point or to determine distances between points, it is necessary to define a Reference System (RS). Maps are the most I common examples. The two-dimensional Cartesian RS is the first defined in Spanish Schools. Two perpendicular axes, XY, and a unit of length are defined. The coordinates of any point are obtained by projecting the position of the point on the axes. The 2D Cartesian System is the easiest to introduce and can be naturally extended into 3D with axes XYZ. The .P(x,y) definition of coordinates through normal y projections is easy: besides, each of the coordinates gives information about the distance to the corresponding axis. For this reason, this reference system is not suitable for defining the location of objects without knowing their distance to the centre of the RS; this is the case of most of x the astronomical sources. Polar Reference Systems (RSs) are the most natural systems to define the position of a point located at an unknown distance. These are natural systems used since childhood to point towards something; the easiest version is to use the ground as a reference plane and mark the location of an object by its “elevation” above that plane and the angle between the line-of-sight to the object and some fiducial direction as, for instance, the corner of the room or the tower of the church. The abstraction of this natural SR is named the Spherical Polar System. The position of a point is given by three coordinates: two angles (φ,θ) and a distance (r) as shown in the figure. The geographical (geographical latitude and longitude) and the astronomical RSs are Spherical Polar systems. The differences between the various astronomical RSs are marked by the spatial orientation of the reference plane (or the direction perpendicular to it also as polar axis in the astronomical context) and the orientation of the fiducial direction in the reference plane. 1 Normal to the reference plane XY z .. P(x,y,z) P(r,θ,φ) y x Fiducial Direction The relationship between the Cartesian and the Spherical Polar coordinates is derived from the figure: x = r sinφ cosθ y = r sinφ sinθ z = r cosφ Once the spherical polar coordinates of two points are known P(r1, θ1, φ1) y Q(r2, θ2, φ2) the distance between them can be calculated by: – Transforming Spherical to Cartesian coordinates and using the expression : 2 2 2 d(,)()()() P Q= x2 − x 1 + y2 − y 1 + z2 − z 1 2 – Using the cosine formula of plane trigonometry and applying it to the triangle OPQ 1 P r1 d(P,Q) φ1 χ Q φ2 r2 O θ2 θ1 The main astronomical RSs are classified by: – the fundamental plane in: alt-azimutals, equatorials, ecliptics... – the location of the centre in: topocentrics, geocentrics, heliocentrics, baricentrics... Fundamental Planes Fundamental planes are defined as follows: Celestial Horizon: plane perpendicular to direction of the Earth gravity field2 at the location of the observer. Celestial Equator:: plane perpendicular to the axis of rotation of the Earth. Ecliptic: plane which contains the orbit of the Earth around the Sun.. The Celestial Horizon depends on the location of the observer on the Earth’s surface, and roughly coincides with the plane tangent to the figure of the Earth3 at the 1 Note that application of this formula requires the angle between r1 and r2 to be calculated,, χ, this can be shown to be, cos χ = cosφ1 cosφ2 + sinφ1 sinφ2 cos(θ 2 −θ1 ) , from spherical trigonometry. 2 This direction is often marked with a plumb and it is also known as the plumb line or the astronomical vertical. 3 The figure of the Earth is close to an ellipsoid of revolution. 3 observer location. The four cardinal points (North-South-East-West) are defined by the projection on the Horizon of the local “parallel” and “meridian” as shown in the figure: Rotation Axis of the Earth N Zenith E W Celestial Equator Centre of the Earth Celestial S Horizon Sky Observer (from a land point surface Zenith S E W N All RSs using the celestial horizon or the zenith to define either the reference plane or the fiducial direction on that plane, are tight to the surface of the Earth, and thus, rotate with it. The apparent motion of the Sun during the day and the stars during the night (east to west) just indicates that we, as observers of the sky, are bound to the surface of a rotating body: the Earth. The “true” motions of the stars are not appreciable at first sight. Thus, it is necessary to define a non-rotating SR to assign coordinates to the astronomical bodies that can be used for their identification. A new reference plane is introduced for this purpose: the ecliptic plane. The intersection between the ecliptic and the celestial 4 equator defines a straight line which is used as a fiducial direction in the non-rotating SRs. This straight line is named the nodal line and points in two well known directions in space: the Aries (γ) and Libra (Ω)4 constellations, marking the equinoxes. Ω Su Nodal line Ecliptic γ Celestial Equator Earth Alt-Azimutal Reference System The XY plane is the celestial horizon at the observer location and the polar axis (axis Z) points to the zenith. On the XY plane, the fiducial direction points to the south and the angles are measured from south to west. The alt-azimutal coordinates are two angles: Azimut (A) and Height (h). The relation between Cartesian and the alt-azimutal coordinates is: 4 In reality, neither the equator nor the ecliptic are fixed. The axis of rotation of the Earth changes direction in space due to the internal structure of the Earth and the gravitational interaction with the Sun, the Moon and other bodies in the Solar system. The orbit of the Earth is also slightly variable because of the action of other bodies in the Solar system. This problem is solved by determining the date or epoch to which the ecliptic and the equator used correspond. 5 Zénith r x = r cos h cosA y = r cos h sinA z = r sin h North h Sou A West (Absolute) Equatorial Reference System The plane XY is the celestial equator (the plane parallel to the equator of the Earth passing through the observer location), and the polar axis (axis Z) points to the North celestial Pole. The fiducial direction in the XY plane is marked by γ and angles are measured in the sense of Earth rotation (towards the East). The absolute equatorial astronomic coordinates are: Right Ascension(α) and Declination(δ). The relationship between the Cartesian coordinates and the equatorial coordinates is: North Pole r x = r cosδ cosα y = r cosδ sinα z = r sinδ δ α γ The Absolute Equatorial system is used, by default, to search and identify the astronomical sources. For instance, all exoplanetary systems in Appendix 1 are identified by one pair (α, δ) and one epoch5, which fixes the orientation of the equinox. These coordinates are used in the application “The smart space traveller” to represent the location of the known planetary systems. 5 The epoch is defined by its Julian date. Standard epochs were 1950.0 and 2000.0; the decimals indicate the fraction of the year passed since the beginning of the years of reference 1950 or 2000.. 6 Notice that the alt-azimutal system is the simplest for an observer situated on Earth (that is, in rotation) while the (absolute) equatorial system is the most suitable to render the coordinates of the astronomical sources; and an additional system is defined in between the two6 that has a fundamental plane, the celestial equator but uses as fiducial direction the “projection of the south” as marked in the figure and, henceforth, it is tight to the Earth rotation.. The angles are measured from the X axis X, in a counter Earth rotation direction (i.e., following the apparent movement of the stars from east to west). The (“hour”) equatorial astronomic coordinates of a star in this system are: Hour Angle (H) and Declination(δ). The relationship between the Cartesian coordinates and the (“hour”) Equatorial coordinates is: North Pole r x = r cosδ cos H y = r cosδ sin H z = r sinδ δ West H Ecliptic System of Reference Plane XY coincides with the ecliptic: the plane containing the orbit of the Earth around the Sun. The polar axis (axis Z) points to the north pole of the ecliptic. On this plane XY, the angles are measured from a point γ toward the East (in the direction of the Earth rotation and orbital motion). The ecliptic astronomical coordinates are given by two angles: Ecliptic Longitude (λ) and Ecliptic Latitude.