Hale COLLAGE (CU ASTR-7500) Topics in Solar Observation Techniques Spring 2016, Part 1 of 3: Off-limb coronagraphy & spectroscopy Lecturer: Prof. Steven R. Cranmer APS Dept., CU Boulder [email protected] http://lasp.colorado.edu/~cranmer/ Lecture 2: Describing the radiation field Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016

Brief overview Goals of Lecture 2: 1. Understand how astronomers define the radiation field 2. Relate that to how physicists discuss electromagnetic radiation Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Energy flux A fundamental concept: how much radiative energy crosses a given area per unit time? Alternately, if one transports a parcel with known

energy density U with a velocity v Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific intensity Often we want to know more the full 3D distribution of photon properties. Specific intensity describes everything contained in the flux, plus how the photons are arranged in direction and in frequency I describes how much photon energy is flowing through a particular area in a particular direction (i.e., through a particular sold angle) per unit frequency (i.e., energy bin)

per unit time Standard units: J / s / m2 / sr / Hz Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific intensity In vacuum, were not considering light rays that bend (no GR!) specific intensity is constant along a given ray (unlike flux) d can mean either into or out of the projected area ^

dA . n = dA cos = cos Alternate units: J / s / m2 / sr / Hz change Joules to photons (divide by E = h) instead of per unit frequency, use wavelength or photon energy bins (conversion: chain rule) Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific intensity In reality, I describes the flux of

energy flowing from one area dA1 into another (dA2). However, since we prefer to specify I locally (all properties at one location), we convert one of the areas into solid angle measured from our location. Both descriptions are identical! Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Angle-moments of specific intensity

Sometimes, I is too much information We can integrate over the spectrum (specific intensity total intensity) We can take weighted moments over the solid angle distribution of rays Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 0th Moment: Mean intensity Just average over all angles: By noting that (dA dt) times c gives a volume, we can compute the mean radiative

energy density (i.e., energy per unit volume), and its proportional to the mean intensity: Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 1st Moment: Radiative energy flux Flux is a vector quantity whose direction gives us ^ the weighted peak n of the angular distribution. Its easiest to think about computing the flux in a particular direction e.g., the z direction:

Thus, Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Higher moments? Useful for stellar atmospheres, but lets skip them for now. We can get a better understanding of all these Is and Js by looking at specific geometries. Lecture 2: Describing the radiation field

Hale COLLAGE, Spring 2016 Specific geometry 1/4: Isotropic I is constant, independent of direction: Example: Planck blackbody equilibrium Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific geometry 2/4: Two-stream I is isotropic in both hemispheres, but up down: I+

(mean) I (net flux = difference) In stars: deep interior .. isotropic ( I+ I ) lower atmosphere .. mostly up, some down ( I+ > I ) upper atmosphere ..

escaping ( I+ >> I ) Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific geometry 3/4: Plane waves In a way, its the exact opposite of an isotropic distribution: I 0 only for one specific direction. In a way, it doesnt matter whether: the radiation field fills all space (like a plane wave),

or is just a narrow beam from a point source the angular distribution, measured from some point inside the beam, is still peaked at a single point in solid-angle-space. Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Specific geometry 4/4: Spherical expansion The usual spherical cartoon focuses on the point of origin: However, if a central sphere is the source of radiation (assume two-stream, I+ 0, I = 0), the observer looks

back to see a uniform-brightness disk on the sky: J H K Far from the source Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 How does classical E&M treat radiation? Lecture 2: Describing the radiation field

Hale COLLAGE, Spring 2016 Electromagnetic waves In vacuum, Maxwells equations become a wave equation, with solutions depending on geometry: Each component of E & B have oscillating solutions, but the only nonzero ones are transverse to k, and to one another: Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Poynting flux Conservation of electromagnetic energy (again, in vacuum) says that if local

energy density U changes at one location, it must be due to an energy flux S into our out of that point: For transverse waves, the time-averaged flux is proportional to the square of the field amplitude S is a true flux: energy density (U) x speed (c) Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016 Plane waves vs. spherical waves Details of the solution depend on the geometry: Cartesian: plane waves have

constant amplitude Spherical: central source, with amplitude ~ 1/r In the spherical case, the simplest central source is an oscillating electric dipole. More complex sources are associated with higher-order E&B multipoles (i.e., antenna theory) For r >> source size, Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016

Next time What happens when the beam passes through matter: radiative transfer What if the beam consists of a superposition of >1 plane waves, each with its own phase and transverse electric field direction? Lecture 2: Describing the radiation field Hale COLLAGE, Spring 2016