AST111, Lecture 1b Measurements of bodies in the solar system (overview continued) Orbital elements
Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends on the semi-major axis and on the planet s mass. Moons or satellites can be used to estimate masses of planets or planetesimals. Perturbations on other planets. Neptune was discovered because it perturbs Uranus. Spacecraft tracking data. Doppler shifts can be measured precisely, allowing direct measurement of gravitational acceleration. Perturbations on nearby debris such as rings.
Planetary Size Actual sizes can be estimated from angular sizes once the distance to an object has been estimated from its orbit. Remember that Kepler s third law relates the orbit period to it semi-major axis. However, angular resolution from most earth ground based observations is limited by atmospheric seeing. (~1 ) Radar echoes. Intensity drops as 1/r 4 so only nearby objects can be studied. Occultations of a bright star. Requires accurate predictions, however gives good size estimates, particularly if more than one occultation of bright stars occur. An occultation is when an object blocks the light from an other object. Using albedos. The albedo is the reflectance or the fraction of light which is reflected from the object. Once you know how far away from the sun the object is, and its distance, you can estimate how bright it is. By comparing brightnesses at different wavelengths it is possible to constrain the uncertainties in the albedo. If you know the albedo accurately then you can estimate the size of the object accurately.
Rotation Measurements: Observe markings on the object s surface and track them in images at different times. If the planet has a magnetic field, charged particles will rotate with the planet. Radio signals from the charged particles should have same periodicity as the rotation of the object. Observing a lightcurve. Some of the surface may have higher or lower albedo, so the amount of sunlight reflected from the object will vary depending upon which side is facing the observer. The rotation period of most objects is between a few hours and a week, with the exception of Mercury and Venus (59 and 243 days, respectively).
Rotation (continued) The angle between the planet s axis of rotation and the vector which is perpendicular to the ecliptic plane, is called the obliquity. The ecliptic plane is the plane containing the planets. Six of the nine planets rotate in the same direction as they orbit the sun. We say they rotate in the prograde direction. They typically have obliquities < 30 o Pluto and Uranus are tilted with rotation axis nearly in the ecliptic plane. Obliquities are ~90 o. Venus has a retrograde rotation, obliquity > 90 o. Venus s obliquity is 177 o. Satellites often rotate at the same period as their orbital period. For example, we always see the same side of the moon because the moon rotates so that the same side always faces the Earth. We say the moon is tidally locked with the Earth.
Shape Measurement: Direct imaging. Stellar occultations. Analysis of radar echoes. Self-gravity tends to produce bodies of spherical shape. Material strength maintains shape irregularities which may be produced by accretion, impacts, or geological processes. Typically bodies which are larger than ~200km are spherical. Mars and Phobos
Temperature The equilibrium temperature can be estimated from the energy balance of that absorbed compared to that emitted. For many bodies the total energy absorbed by the sun balances that emitted thermally. However, many planets also have internal heat sources. There can be diurnal (daily), latitudinal and seasonal variations in temperature. Radiation transfer through the atmosphere can be complex (such as happens where there is a greenhouse effect). Measurements can be taken in situ or by spectroscopic study of the atmosphere and surface.
Magnetic field Magnetic fields are produced by moving charges. Currents moving through a solid usually decay quickly. Large planetary magnetic fields are produced by a dynamo which requires a moving fluid core. The giant planets, Earth and Mercury have magnetic fields generated from their interiors. Smaller ones can be produced by remnant ferromagnetism. Mars and the Moon have localized crustal fields. Magnetic fields can also be produced by the interaction between the object s ionosphere and the solar wind. (Venus and comets).
Atmosphere composition and structure The planets and some satellites are surrounded by gaseous atmospheres. These are measured spectroscopically at a variety of wavelengths. Occultations also give information about the vertical structure. Gas giants tend to have mostly H 2, He. Venus has a thick CO 2 atmosphere. Earth s atmosphere is nitrogen and oxygen. Mars has a tenuous CO 2 atmosphere. Titan has a dense nitrogen atmosphere containing many organics. Pluto and Triton have tenuous nitrogen atmospheres. Mercury and the moon have extremely tenuous atmospheres of atomic atoms (Mercury) and He and Ar (the moon).
Bulk Composition The bulk composition must be inferred indirectly. Constraints include the size and mass of the object, structure and composition of the atmosphere and the observed effective temperature. Models as a function of radius can be constructed which attempt to match these constraints. Gravitational, magnetic and seismic measurements can also provide constraints.
Orbits of the planets and asteroids Inner solar system Outer solar system 12
Orbits and Spins of Planets 13
Orbits: Kepler s laws All planets move along elliptical paths with the Sun at one focus. A line segment connecting the planet and the Sun sweeps out area at a constant rate. The square of the planet s orbital period about the Sun P orb is proportional to the cube of its semi-major axis, a. 2 3 P a orb Kepler s laws can be derived from Newton s laws of motion and gravity in the case of two massive bodies under the sole influence of gravity (we will do this next week). 14
Orbital elements An orbit of a massless (very low mass) particle about the Sun in 3D can be specified by 3 components of the velocity and 3 components of the position at a particular time. This is a Cartesian system. Alternatively we can describe the orbit in terms of three parameters which determine the shape and orientation of the orbit and three angles which determines where the particle is along the orbit again at a particular time. These six parameters are known as orbital elements. a e i semi-major axis eccentricity inclination f ω Ω true anomaly argument of periapse longitude of ascending node 15
Orbital elements (continued) 16
xx Why the difference in nomenclature?
For the two body problem, closed orbits are elliptical. Periapse or perihelion is the point of closest approach to the Sun or central object. Apoapse or aphelion is the most distant point from the central object. The Sun lies at a focal point 18
Solar eclipses Moon s orbit is inclined with respect to the ecliptic Longitude of the ascending node for the moon s orbit lines up with the Earth- Sun axis during an eclipse Moon s orbit precesses fairly quickly. Eclipses happens about twice a year.
Orbital elements (continued) Principal orbital elements a describes the size of the orbit e describes the shape i describes the tilt (inclination) The angles that go with each of them f (true anomaly) describes the location of object in the orbit. ϖ (longitude of periapse) or ω (argument of perihelion) describe orientation of perihelion Ω (longitude of the ascending node) describes position where orbit crosses the plane of the ecliptic. 20
Orbital elements (continued) Principal orbital elements a units of length positive for an elliptical orbit, infinity for a parabolic orbit, negative for a hyperbolic orbit. e unitless ranges from 0 to 1 for an elliptical orbit. Is equal to 1 for a parabolic orbit. Greater than 1 for a hyperbolic orbit. i degrees or radians ranging from 0 to 180 o (> 90 o retrograde) The angles f ϖ Ω degrees or radians ranging from 0 to 360 o 21
Nomenclature on angles angles are called arguments angles given with respect to perihelion or periapse are called anomalies angles given with respect to the vernal equinox (or line of sight) are called longitudes. Slow variations are called secular variations. Precession of the orbit is secular. Both ω and Ω can precess 22
Orbital elements (continued) Of the 6 orbital elements, 5 are slowly varying, (constant when the system is 2 spherical bodies alone under the sole force of gravitational attraction). The true anomaly f, (or mean longitude M), is the only element which varies quickly in time. Lots of conserved quantities! Perturbation theory (from purely Keplerian motion) is appropriate 23
The orbit r where f is the angle called 1 1 ecos f the true anomaly. Describes the ellipse radius from the Sun e is the eccentricity You can figure out the constant of proportionality in terms of the semi-major axis and eccentricity 24
The orbit What values of true anomaly give the maximum and minimum radii? r 1 1 + e cos f What is maximum and minimum radius in terms of the semimajor axis?
The orbit r 1 1 + e cos f What values of true anomaly give the maximum and minimum radii? What is maximum and minimum radius in terms of the semi-major axis? ANSWERS: rmax = A/(1-e) at f=0 rmin = A/(1+e) at f=π r = A at f=±π/2 r = A 1+e cos f
Low eccentricity approximation Eccentricity 0<e<1 (though unbound objects can be described with e>1) 1 1+x 1 x +... 1 1 x 1+x +... Taylor series for small x rmax = A/(1-e) is approximately equal to A(1+e) at low eccentricity (nearly circular orbit)
rmax = A/(1-e) at f=0 rmin = A/(1+e) at f=π r = A at f=±π/2 Semi-major axis a 2a = rmax + rmin = 2A/(1-e 2 ) A = a(1-e 2 ) Semi-minor axis b However b A because center of ellipse is not at r=0
Who keeps track of orbits of planets and planetsimals? Up to date information on web sites provided by the Minor Planet Center and JPL s Solar System Dynamics Group. MPL is a good site to get locations of comets and asteroids. If you find a new object you report your observations to this site and they help you calculate the orbital elements. This allows others to find the object again later. Minor Planet Electronic Circulars Circulars (MPECs) are such reports. A list of positions on the sky (in RA and DEC) as a function of time is known as an ephemeris. Many objects are being discovered as part of active on-going searches (satellites of giant planets, Kuiper Belt objects, comets and small asteroids). Orbital elements of recently discovered objects may not be very well measured. Measurements over years are often required before the elements are known to any precision. Spacecraft missions in order to be successful require accurate predictions for orbits (e.g., think about the Cassini mission) 29