Lab 3: Astrometry

Let’s calculate the proper motion of the “26 ” (discovered in 1853) Lab 3: Astrometry

Let’s calculate the proper motion of the asteroid “26 Proserpina” (discovered in 1853).

What do we need to measure a proper motion of an asteroid?

What kind of analyses do we need to conduct? Lab 3: Astrometry Let’s calculate the proper motion of the asteroid “26 Proserpina” (discovered in 1853).

• Multi- images of the asteroid that can tell us the changes in the asteroid sky positions (web page; use NGC 7331 for training purpose); • Measurements of the asteroid positions on CCD images (centroid locations); • Mapping between the CCD coordinates and sky coordinates (linear algebra matrix conversion); • Precise measurements of the asteroid sky positions (using known sky positions of nearby stars); • Proper motion measurements in unit of angular distance per unit time (e.g. arcsec/hour) The Celestial Sphere and Sky Coordinates The Celestial Sphere: North and South Poles

The celestial equator is an extension of the Earth’s equator to the surface of the celestial sphere

The north/south celestial pole is an extension of the Earth’s north/south pole. The Celestial Sphere: Earth’s Latitude & Longitude Two perpendicular great circles: the equator & Greenwich longitude circle Toronto: 43.7N (latitude) & 79.4W (longitude) Note: 1 hour is 15 degrees in longitude and Toronto is 5 (NRAOhours is located behind London at Charlottesville.) The Equatorial Coordinate: (R.A. & decl.) • Right Ascension (R.A.): extension of the longitude to the celestial sphere • Declination (DEC): extension of the latitude to the celestial sphere (vernal equinox) The Equatorial Coordinate: (R.A. & decl.) (R.A., decl.) and Vernal Equinox Vernal equinox is the ascending node of the on the ecliptic plane when it meets with the celestial equator.

Ecliptic Plane represents the motion of the Sun and solar system objects. (R.A., decl.) and Vernal Equinox Equinoxes are where the Sun meets the celestial equator (Vernal equinox for spring; Autumn equinox for autumn)

Motion of the Sun through the year on Vernal Equinox: the ecliptic plane. ascending node of the ecliptic on the equator (R.A., decl.) and Vernal Equinox Also we need to understand sexagesimal coordinate system in (R.A., decl.).

One second in R.A. is how many The Sun Right Declination arcsec in decl.? Ascension Vernal Equinox 0h 0 Summer Solstice +6h +23.5 Autumn Equinox +12h 0 Winter Solstice +18h 23.5 Motion of the Solar System Objects Gravitational Motion and Orbit

Planets orbit around the Sun. (Note that most of the in the Solar System belongs to the Sun.)

What are their orbits? And how about other objects in the Solar system? How are their orbits related to the escape velocities? Gravitational Motion and Orbit

Depending on the escape velocity (or the total energy), the orbiting object can take different paths of conic sections:

o v < vE → Elliptical Orbit (including Circular Orbit)

o v = vE → Parabolic Orbit o v > vE → Hyperbolic Orbit v = escape velocity, Potential orbits of the planet P around E a star at the center (filed circle). vc = circular velocity Gravitational Motion and Orbit

The circular or parabolic orbit is difficult to achieve because they requires a special condition that is difficult to achieve: 2 vC = Gm/r (circular velocity) 2 vE = 2Gm/r (escape velocity).

Therefore, the two general orbital solutions are: o Elliptical Orbit (bound case for negative total energy) o Hyperbolic Orbit (unbound case for positive total energy) Potential orbits of the planet P around a star at the center (filed circle). In the Solar System, the planets follow elliptical orbit. Do we have any object with hyperbolic orbit? Gravitational Motion and Orbit

“ have elliptical or parabolic/hyperbolic orbits”

Comets of hyperbolic orbits will leave the Solar system at the end (cf: non-periodic comets)

Orbits of Kohoutek (red) and Earth (blue), illustrating the high eccentricity of the orbit and more rapid motion when closer to the Sun. Solar System Objects in Elliptical Orbit

Planetary orbits are inclined. How many parameters (= orbital elements) do we need to describe a planetary orbit with respect to a reference frame?

Planetary orbital plane Reference plane (e.g., Equatorial)

Planetary orbital plane Reference plane (e.g., Equatorial) How many parameters (= orbital elements) do we need to describe a planetary orbit with respect to a reference frame?

Planetary orbital plane Reference plane (e.g., Equatorial) 1. Size & Shape of Elliptical Orbit 2. Orbit orientation (tilt, twist, rotation) 3. Reference position (or time). Planetary orbital plane Reference plane (e.g., Equatorial) How many parameters (= orbital elements) do we need to describe a planetary orbit with respect to a reference frame?

Vernal point

Ascending node Orbital Elements

Vernal point Ascending node

1. Size & Shape: Semi-major axis (a) and eccentricity (e). 2. Orbit orientation: Tilt (inclination, i), Twist (longitude of ascending node, ), Rotation (argument of perihelion, ) [ 3 angles;  and  are measured in the counter-clockwise direction and show the location of the node and rotation of the plane, respectively] 3. Reference position (= time): Epoch of perihelion (). Determination of Center of Stars, Telescope Plate Scale, Mapping between the CCD and Sky Coordinate Why do bright stars “look” bigger than faint stars?

Telescope Plate Scale

Plate scale is (usually) defined as the number of arcseconds per mm (or cm) at the focal plane.

Sometimes it means the number of arcseconds that the single pixel of the detector (e.g., CCD) spans. Note that the detector is located at the focal plane. Telescope Plate Scale (= Image Scale) Plate Scale: the relation between an angular distance on the sky and a physical distance in the telescopic focal plane

•D: diameter of the primary •f: focal length of telescope •f/# (focal ratio) = f/D •u: angular distance on the sky in arcsec() •s: linear distance at the focal plane s = f x u = (f/#) x D x u()/206265  If the focal length is 100 m, it gives a plate scale of  2/mm (or 0.5 mm/) Note: 1 radian = 206265, tan u  u if u << 1 (u in radian)

(Example) The Palomar 5 meter telescope has the focal ratio of f/16. Using the above equation, we can obtain the plate scale of 2.56/mm (or 0.388 mm/). If its CCD has 2048 pixels and each pixel is 18 micron size, the field of view is 1.8 arcminute.