**Table of Contents**

Last updated: Feb 18, 2011

Given HIFI's high spectral resolution, a relativistic treatment is necessary when accounting for spacecraft motion. Lorentz transformations take observed frequencies from one inertial frame to another. Over the ten seconds or so of a single integration, we can ignore the spacecraft acceleration. We can likewise ignore General Relativistic (GR) effects due to Herschel's elliptical orbit; and we choose to ignore GR effects when observing near interfering bodies such as Jupiter.

The internal frequency calibration schemes for HRS and WBS, executed within the HIFI Level 0.5 pipeline, produce observed frequencies in the spacecraft frame ("HSO"). They are in the IF scale; the frequency of the detected photon is simply IF + LO. There are two observables of interest: the frequency of the incident wave, and its direction. In fact, we don't know the direction from which the radiation was incident because the HIFI beam is of finite size (between 10 and 45 arcseconds FWHM), and additionally has a small pointing uncertainty (about 2 arcseconds). Often the signal will come from a resolved source. We assume the direction of incidence is the boresight of the beam as reconstructed in the pointing product; the frequency error due to the uncertainty in signal arrival direction is small, of order 1 kHz per arcsecond of error. Note that because of stellar aberration, the direction of incidence in the HSO frame differs from that observed in, say, the Solar System Barycenter (SSBC) frame, by up to about 23 arcseconds. In practice, the pointing information provided in the pointing product has been de-aberrated, and so is in the SSBC frame.

Only the frequency remains known in the HSO frame alone; everything else, including the HSO motion, is known in the SSBC frame.

The Solar System Barycenter is the fundamental inertial frame for calculations involving the motion of HSO. The state vectors (r,v) of solar system objects (SSOs), including HSO, expressed in this frame have as origin the SSBC and as reference directions the International Celestial Reference Frame axes.

The motion of HSO with respect to the Geocenter is determined to an accuracy of about 5 cm/s by the usual tracking techniques. The Geocenter is tied to the Solar System Barycenter through the JPL DE405 planetary ephemerides to a precision of mm/s. Because the HSO motion, target coordinates, and telescope pointing are all defined in this frame, the transformation from the HSO frame to any other passes necessarily through the SSBC.

It's important to keep in mind in which frame observables are defined; in the equations below we use the convention that subscripts refer to the object of interest, and superscripts to the frame in which the observable is measured. For example, the direction of a SSO as seen by the telescope is

and as seen by an observer at rest in the SSBC, it is

A signal is incident upon the spacecraft and detected at frequency

The transformation of HSO-centric frequencies to SSB-centric is described by the relativistic Doppler formula:

where

and is the de-aberrated direction of telescope pointing (the J2000 coordinates of the beam boresight at the time of observation).

It might be useful to review the definition of the Local Standard of Rest. Take any point in the Galactic plane, and imagine there exists a circular orbit about the Galactic Center that passes through that point. The circular velocity defines the Local Standard of Rest for that position. Such a point coincident with the Sun defines the Solar Local Standard of Rest (LSR). The Sun has a peculiar velocity with respect to the LSR, which can be estimated in different ways. The LSR so defined is also called the Dynamic LSR, referring as it does to the rotation curve of the Galaxy. In practice, the Sun's peculiar motion with respect to the LSR has also been inferred from the mean motion of bright stars in catalogues or in the solar neighborhood. The LSR defined by this calculation of peculiar motion is called the kinematic LSR (LSRk), and is the more commonly used convention. However, there is not a single standard value for the LSRk. Further, it is not very close to any physical velocity of interest; it is only a common convention.

We take as our definition of the LSRk frame: the motion of the SSBC with respect to the LSRk is 20.0 km/s toward ra, dec = 18h03m50.29s, +30:00:16.8 (J2000). This is a common observatory standard, and adopted by many astronomical software suites such as CASA, SLALIB, and CLASS.

Frequencies in the LSR frame are derived from the SSB frame by a Lorentz transform:

And using the relativistic Doppler formula, LSR frequencies can be calculated directly from observed HSO frequencies and SSB-centric known quantities:

Because the 3-velocity of a star or other such target is unknown, a transformation to its comoving frame is impossible. What may be known about the object is that a spectral line appears shifted from its expected rest frequency, and that shift can be interpreted as a radial velocity. One would like to see the relative velocities of other spectral lines, with a view to constraining a dynamical model of the object. In this context, transforming frequencies to the source frame is only a shift of the frequency axis already defined in an inertial frame (e.g. SSB, LSR), and then expressing the frequencies as velocities according to some convention. There are three operative conventions for expressing redshift as a velocity:

Convention | ||
---|---|---|

radio | ||

optical | ||

relativstic |

The relativistic definition would be correct if the relative velocity between observer and source were purely radial. The radio and optical definitions are two linearisations of the relativistic equation, and their difference is quite large at HIFI bandwidth about 4 GHz and frequencies 500 GHz

The HIFI Pipeline task `DoVelocityCorrection`

will, if requested, recast the frequencies in the data to
the ``source frame'' of a nonSSO target, but it is not a real frame change and a Lorentz transform is not performed.

It is often interesting to view spectra plotted on an axis representing speed. In recent versions of HIPE (6.0.1360 or earlier),
the Spectrum Explorer display tool uses the relativistic convention when displaying frequencies as velocties; however, the
`ConvertFrequencyTask`

uses the radio definition.

When observing objects within the solar system, we know or can estimate their full 3-velocity, and so a real Lorentz transform to the object rest frame is performed. The manner is thus: the state (r,v) of Herschel is known as a function of time, as is the SSO's. A photon received by HSO at time t was emitted by the SSO at time t - LT(t), where LT(t) is the light-travel-time between the two. The frame to which we transform is the comoving frame of the SSO at the retarded time t - LT(t).

LT is computed iteratively; three iterations are sufficient to achieve a precision of less than a millisecond within the orbit of Pluto, which is less than the error due to ignoring General Relativity. Once the SSO state at the retarded time is known, frequencies are transformed by equation 2 (with SSO substituted for LSR).