Conceptual and theoretical background of Shape

The Doppler-effect is the change of the the frequency of a propagating wave as detected by an observer that is moving relative to the source of the wave. Without it astrophysics would be severely handicaped. Probably the most fundamental discovery made by measuring velocities based on the Doppler-effect is that of the expansion of the universe by Edwin Hubble as a correlation between the distance of galaxies with the shifted frequencies of spectral lines in distant galaxies.

The change in frequency can be observed in any type of waves, surface waves in water, sound or electromagnetic waves like radio or light. The amount of change depends on the relative velocities between the source, the propagating medium and the receiver. Police uses the Doppler-effect to measure the speed of cars on the road by bouncing short radio waves off the vehicles and measuring the change in frequency of the reflected signal. For further information on the Doppler-effect outside the context of its application in Shape see e.g. the article on Wikipedia and references therein.

In many astrophysical objects there is direct correlation between position and velocity. Therefore, if we can measure the velocity along the line of sight, we can hope to disentangle the information of position that appears flattened in images. I

n addition to images, in Shape the Doppler-effect is a key measurable piece of information that helps reconstruct the 3D structure of many object types using position-velocity diagrams or channel maps.

Spectral absorption or emission lines are shifted when the observer and the emitter are moving relativ to each other. When the lines move towards longer wavelengths we speed of "red shift" (receding), otherwise it is of course "blue shift" (approaching).

In many astrophysical environments there is a quite well defined relation between the velocity vector field and the position vectors. Examples are the surfaces of rotating stars or accretion disk governed by gravity. In both types of objects the rotation determines the direction of the vectors and in the accretion disk gravity provides the magnitude as a function of distance.

In explosive events, like a supernova or ejecta from other stellar events, the gas motion is often radial, just like the position vector with respect to the star. Furthermore, if the ejection is ballistic and from a short event, then the material soon will be sorted according to its speed: the further away, the faster it moves, resulting in a proportionality between the position and velocity vectors.

If sufficient constraints exist, such relationsships between position and velocity vectors can be applied to reconstruct the 3D structure of an object by measuring the projected position vectors in an image and deriving the position along the line of sight from the Doppler-shift in sprectroscopic observations.

In addition to images, spectroscopic observations of the velocity along the line of sight are one of the key elements for 3D reconstructions using Shape. The first type of spectroscopic information that was produced in Shape to be directly compared with observations where position-velocity (p-v) diagrams (see the illustrations on the right). P-V diagrams are still the most common type of observation for 3D reconstructions in Shape.

Channel maps

A channel map is an image that contains only emission from within a given interval of velocities along the line of sight, i.e. a certain velocity "channel".

Channel maps are the most common form of displaying velocity information in molecular radio observations. A "stack" of channel maps is called a "data cube", with two spatial dimensions and one velocity dimension.

In Shape you can use channel maps a in a similar way as position-velocity diagrams to reconstruct the 3D structure of an object, if the mapping between the velocity and spatial component can be derived. If this mapping is linear, then each channel map represents a "slice" through the object. While a p-v diagram is a slice along the line of sight, a channel map is a slice in the plane of the sky.

At optical wavelengths channel map are the natural representation of the results from Fabry-Perot observations. A Fabry-Perot etalon is basically a tunable filter, through which passes only a very narrow range of wavelengths, i.e. velocities. It allows the observer to step through a limited spectral range and record adjacent velocity "channels".

Lightechoes

Light echoes occur when the emission from a highly variable light source is reflected of the dust or gas around the source and the reflection changes with a time difference that is larger that the time scale of the variability. It then produces an expanding reflection pattern in the nebula around the light source.

In a reflecting telescope the mirror usually has a parabolic shape, because it collects light coming from far away in one point, the focus. Furthermore, all rays are synchronized, i.e. "in phase", since the time they it takes them to travel from the object to the focus is the same for all. In a light echo, the light path is reversed. Leaving from the star (focus), at any given time we detect those reflections that come from a parabola within the dust nebula. As time goes on, the parabola from which we receive light changes shape and sweeps through the nebula. In principle, continuous observations of the light echo allows to reconstruct the 3D structure of the nebula.

For now see: modifiers

Radiation Transport

See the PDF document on the "Physics in Shape" for information on radiation transport basics and how it is applied and approximated in the Shape.