1B11 Foundations of Astronomy Orbits

1. 1B11 Foundations of AstronomyOrbits Liz Puchnarewicz emp@mssl.ucl.ac.uk www.ucl.ac.uk/webct www.mssl.ucl.ac.uk/

2. conjunction (full) orbit of superior planet superior conjunction (full) Earth’s orbit inferior conjunction (new) orbit of inferior planet opposition (full) 1B11 Orbits Before we begin our review of the Solar System, this section introduces the basics of orbits.

3. 1B11 Sidereal Period The sidereal period is the time taken for a planet to complete one orbit with respect to the stars.

4. P2 orbit of superior planet P4 P3 P3 P2 P1 1B11 Synodic period The synodic period is the time taken for a planet to return to the same position relative to the Sun, as seen from the Earth. Earth’s orbit

5. 1B11 Kepler’s Laws • The orbit of a planet is an ellipse with the Sun at one focus (1609). • The radius vector joining the planet to the Sun sweeps out equal areas in equal times (1609). • The squares of the orbital periods of the planets are proportional to the cubes of the semi-major axes of their orbits (1619). Johannes Kepler (1571-1630)

6. 1B11 Ellipses a = semi-major axis b = semi-minor axis e = eccentricity q = “true anomaly” r1 r a q F ae b Equation of an ellipse: r + r1 = constant = 2a The eccentricity: and the relation between r and q:

7. 1B11 Kepler’s First Law The orbit of a planet is an ellipse with the Sun at one focus. minor axis perihelion aphelion major axis F2 F1

8. 1B11 Kepler’s Second Law The radius vector joining the planet to the Sun sweeps out equal areas in equal times. At perihelion, the planet moves at its fastest C D B At aphelion, it travels at its most slow A

9. 1B11 Kepler’s Third Law The squares of the orbital periods of the planets are proportional to the cubes of the semi-major axes of their orbits.

10. 1B11 Newton and Kepler centre of mass rSun rEarth The Sun and the Earth rotate about each other, around their common centre of gravity. rSun + rEarth = a Their centrifugal forces must be balanced:

11. 1B11 Newton and Kepler The velocity v may also be written in terms of the radius r and period T: Substituting: Which leaves:

12. 1B11 Newton and Kepler a = rSun + rEarth, so rEarth = a – rSun, and : So: And:

13. 1B11 Newton and Kepler

14. 1B11 Newton and Kepler And finally: which is Newton’s form of Kepler’s Third Law. Notice that the “constant” isn’t strictly constant for every planet, because each planet’s mass will be different. But since the mass of the Sun is so large, it is true to first order.

15. 1B11 Kepler’s Second Law A quick reminder… At perihelion, the planet moves at its fastest C D B At aphelion, it travels at its most slow A

16. 1B11 Orbits P v vt F Dq Q Planet moves from P to Q in time Dt through angle Dq. v = orbital velocity at P vt = transverse component of v DFPQ has area DA where DA = ½ r (vtDt) and DA/Dt = ½ vtr (assuming the ellipticity e is low, ie it’s almost a circle)

17. 1B11 Orbits So since DA/Dt = ½ vtr, as Dt -> 0, dA/dt = ½ vtr But vt ~ rdq/dt = rw , where w is the angular velocity - so dA/dt = ½ r2w Moment of inertia, I = mr2 = r2 (for unit mass) dA/dt = ½ Iw = ½H Where H is the angular momentum per unit mass. Since H is conserved dA/dt = constant ie Keplers 2nd Law

18. 1B11 Orbits Now: So integrating over the orbit: Therefore: Since: We have:

19. 1B11 Orbits At perihelion: where therefore therefore and

20. 1B11 Orbits Similarly, for aphelion: For the Earth, a = 1AU = 1.496 x 108 km P = 1 year = 3.156 x 107 seconds e = 0.0167 Therefore vperi = 30.3 km/s and vap = 29.3 km/s

21. 1B11 Masses from orbits For a body (eg a moon) in orbit around a much larger body (a planet), if you know the period of rotation of the moon, T, and its distance from the planet, a, you can calculate the mass of the planet from Newton’s version of Kepler’s Third Law. Mmoon = mass of the moon Mplanet = mass of the planet, and Mplanet >> Mmoon G = Gravitational constant So then: P2 = 4p2/GMplanet x a3

22. 1B11 Masses of stars in binary systems In visual binary stars, we can sometimes observe P and measure a if the distance to the binary is known. We can then solve for the sum of the masses, ie: (m1 + m2) = (4p2/G) + a3/P2 (P is typically tens of thousands of years) If the stars have a high proper motion, the centre of mass moves in a straight line and a1 and a2 can be measured. m1r1 = m2r2 In a few cases, can solve for m1 and m2.

23. 1B11 Masses of stars spectroscopic binaries Spectroscopic binaries are those binary systems which are identified by periodic red and blue shifts of spectral lines. In general, the parameter (m1 + m2) can be calculated. Sometimes the individual masses can be calculated.

24. 1B11 Eclipses Eclipses occur when one body passes directly in front of the line of sight from the observer to a second body. For example, a solar eclipse absolutely not draw to scale!

25. 1B11 Solar eclipses Important facts: The Moon’s orbit is inclined to the ecliptic by 5.2O, so an eclipse will only occur when the Moon is in the ecliptic plane. The angular diameter of the Moon (which varies between 29.5 and 32.9arcmins) is very similar to that of the Sun (32 arcmins), which is why solar eclipses are so spectacular. There are three types of eclipse – Partial – the observer lies close to, but not on, the path of totality Annular – the Moon is relatively distant from the Earth

26. 1B11 Three types of eclipse There are three types of eclipse – Partial – the observer lies close to, but not on, the path of totality Annular – the Moon is relatively distant from the Earth, so a ring of Sun appears around the Moon’s shadow. Total – when the Moon’s and the Sun’s angular diameters match. At the point of totality, the Sun’s corona (its outer atmosphere) appears.

27. 1B11 Lunar eclipses When the Earth lies directly between the Sun and the Moon, a lunar eclipse occurs. From the Earth, we watch as the Earth’s shadow passes across the face of the Full Moon. As seen from the Moon, the Earth has an angular diameter of 1O 22’, so there are no annular lunar eclipses. The Earth’s shadow is not black however, light from the Earth’s atmosphere reaches the Moon during totality and we see this light reflected from the Moon. This light is red – the blue light has been scattered away by dust in the atmosphere. In a typical lifetime, you should see about 50 lunar eclipses from any one location – solar eclipses are much more rare.

28. 1B11 Eclipsing stars If the orbital plane of a binary system lies close to, or along, our line of sight, then we will see changes in the lightcurve as the eclipses occur. period flux secondary eclipse primary eclipse time

29. 1B11 Transits A transit is when a small body passes in front of a much larger one. We can observe transits of Mercury and Venus across our Sun, for example. We also search for evidence of transits by extrasolar planets, passing in front of their local stars. The drop in flux is tiny, but measurable if the relative angular size of the planet is large enough, eg a Jupiter-like planet in close orbit (Mercury-ish). For planets in our Solar System which have their own moons, eg Jupiter, we can also observe transits as a moon passes across their face.

30. 1B11 Occultations When one object completely obscures another, this is known as an occultation. So when the angular size of the Moon is equal to or larger than the Sun’s, the total solar eclipse is an occultation. Stars are occulted by the Moon or by planets and asteroids. Lunar occultations occur at predictable times so can provide precise positions. [Strictly speaking, an eclipse occurs when one body passes through the shadow of another.]

31. N N S S 1B11 Lunar libration The Moon rotates on its axis once a month, therefore it always keeps the same face pointed towards the Earth. Well almost – the Moon’s orbit is elliptical and inclined to the ecliptic, so we do see “around” the Moon making more than 50% of its face visible in total. 5.2O ecliptic Libration occurs in longitude and latitude and adds up to a “wobble” of about 6O. It’s also called “phase-locking”. Moon’s orbit

32. 1B11 The Solar System - G2V star The Sun Mercury Venus Earth Mars [Asteroid Belt] Jupiter Saturn Uranus Neptune Pluto Comets Terrestrial planets Giant (gaseous) planets and moons Icy Planetessimals