# Theory/X-ray trigonometric parallax

In visual astronomy the distance to nearby stars is calculated using the trigonometric parallax of their movements relative to background stars or galaxies that are immobile within the resolution of the telescope used. When X-ray astronomy detectors have sufficient resolution, it should be possible to measure the X-ray trigonometric parallax of nearby stars.

## Astronomic distances

Distance measurement by parallax is a special case of the principle of triangulation, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 arcsecond,[1] leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see derivation below), the distance to an object (measured in parsecs) is the reciprocal of the parallax (measured in arcseconds): ${\displaystyle d(\mathrm {pc} )=1/p(\mathrm {arcsec} ).}$  For example, the distance to Proxima Centauri is 1/0.7687=1.3009 parsecs (4.243 ly).[2]

Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.[3] Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object."[4]

## Planetary sciences

In theoretical astronomy, whether the Earth moves or not, serving as a fixed point with which to measure movements by objects or entities, or there is a solar system with the Sun near its center, is a matter of simplicity and calculational accuracy. Copernicus's theory provided a strikingly simple explanation for the apparent retrograde motions of the planets—namely as parallactic displacements resulting from the Earth's motion around the Sun—an important consideration in Johannes Kepler's conviction that the theory was substantially correct.[5] "[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy"[5] with the Earth at the center. Using a computer, this means that for competing programs, one written for each theory, the heliocentric program finishes first (for a mutually specified high degree of accuracy).

Orbits come in many shapes and motions. The simplest forms are a circle or an ellipse.

## S stars

Stars "of spectral type S are characterized by unusual photospheric abundances which imply enrichment of the stellar surface by nucleosynthesis products. Spectroscopically, S stars are identified by bands of ZrO and LaO, replacing the TiO bands found in M stars. The spectra of S stars indicate strong enhancement of s-process elements in the photosphere (an accident of nomenclature - when the S spectral type was introduced, the slow neutron capture process was unknown). Abundance analyses show that in S stars, the C/O ratio is very close to unity [...], which also implies the presence of the products of nucleosynthesis at the stellar surface."[6]

The "extrinsic S stars, includes stars which have elemental abundances which appear to have been altered by mass transfer from a binary companion."[6]

The "intrinsic S stars, includes stars which have high luminosity and lie on the asymptotic giant branch (AGB). They show evidence that their compositional abnormalities are a result of nucleosynthesis and [perhaps] convective mixing to the surface. In particular, a defining characteristic which distinguishes the two types is that the intrinsic S stars contain technetium, while the extrinsic S stars do not."[6]

"Both HCN and SiO have readily observable lines at 3 mm."[6] χ Cyg is at a distance of 170 pc, but parallax measurements put it at D = 144 ± 25 pc (Stein 1991), parallax of 5.53 mas (198 ± 38) pc as of 2007 according to SIMBAD.

"All of these stars are bright at 2 µm and therefore have circumstellar dust [...] In one observing session, we obtained a 5 x 5 cross at HPBW spacings for the star χ Cyg in the CO J = 2-1 line. The data were relatively noisy because of limited integration time and weather conditions but do indicate that the envelope is extended with respect to the 25" telescope beam."[6]

χ Cyg was "detected in the SiO v = 1 J = 2-1 maser emission line [...] χ Cyg has an unusually large dust/gas ratio of 9.0 x 10-3 [The dust-to-gas ratio for S stars detected in CO J = 1-0 is] 9.0 x 10-3 [...] For one star in our sample namely χ Cyg, the SiO J = 2-1, v = 0 emission has been mapped interferometrically [...] the SiO abundance at the base of the expanding envelope must be ~ 2 x 10-5 to explain the observed intensity distribution of the SiO emission. Thus, a substantial fraction (30%-50%) of all silicon atoms are in the form of gas phase SiO at the point where molecules are injected into the stellar wind. As the gas moves away from the star, the SiO is depleted from the gas, presumably by the process of grain formation, such that at radii of several x 1015 cm, the SiO gas phase abundance has fallen by > 90%. [...] χ Cyg, which has a relatively low mass-loss rate and hence low envelope opacity to UV photons."[6]

## Theoretical trigonometric parallax

The distance from a parallax measurement in pc is given by D = 1/p in arcseconds. The "standard deviation, which describes the 'typical' amount of error, is not negligible in comparison with the parallax. [...] the non-linearity of the relation causes the distribution function of the estimates [...] and thus the distribution function of the errors in distance [...] to be skewed."[7]

For example, the most recent parallax for χ Cyg is 5.53 ± 1.10. The skewness results when the distance is calculated D = 1/0.00553 = 181 pc, but this ranges from 151 to 226 or 181 +45 and -30 pc.

"Extreme values are less likely to appear in a small sample than in a larger one. The reason is that with a small sample it is not very probable that any of the sample parallaxes will have an absolute value close to zero. As the sample size is increased the absolute value nearest to zero will continually decrease and the corresponding very large value of [D] will drive up the sample variance. Furthermore, this anomalous value will more and more seriously distort the value of the sample mean. The increasingly erratic behaviour of the sample mean as sample size is increased contrasts with the behaviour of the mean for a Gaussian distribution, which more and more closely approaches the population mean with increasing sample size."[7]

## Entities

"Physical entities are (almost) always characterized by a numerical parameter whose value can only be approached through some measuring process, by means of a statistical adjustment of a set of appropriate measurements. All that we can ever obtain is a best (in some prescribed sense) estimate of that value. By itself this estimate is, strictly speaking, meaningless, for it simply defines an interior point of a set of possible values; unless one has some estimate of the extremes of that set (to within some level of probability) in addition to the best estimate itself, one cannot exclude any value of that parameter. In many cases the probabilities of nearby values can be described by a function that is symmetric around the best estimate, and the parameter values bounding an interval (or a region if one happens to be dealing with multivariate analysis) which has a certain cumulative probability (usually 0.68) define an 'error bar' or error interval whose endpoints are equally distant from the best estimate."[7]

## Sources

The "positions of H2O masers in Sgr B2, a massive star forming region in the Galactic center, relative to an extragalactic radio source with the Very Long Baseline Array [have been determined]."[8]

"Most luminosity and many mass estimates (e.g. based on column densities) scale as the square of the source distance, while masses based on total densities or orbit fitting (using proper motions) scale as the cube of distance."[8]

"Most classical techniques involve determining the distributions of large numbers of bright sources that are assumed to be symmetrically distributed about the Galactic center. Distances to these sources are mostly photometric, which require accurate knowledge of absolute magnitudes and detailed calibration of the effects of metallicity, extinction and crowding, and combined systematic uncertainties arguably are at least 5% and possibly larger."[8]

"For both Sgr B2M and Sgr B2N we used the background source, J1745–2820, as a position reference for the parallax and proper motion measurements [...] J1745–2820 is a well-studied extragalactic radio source (Bower, Backer & Sramek 2001) that has been used in previous VLBA astrometric observations of Sgr A* [...] J1745–2820 is projected only 20′ west of the maser sources, making it a nearly ideal position reference as it is very close to our maser targets, thereby canceling most systematic errors by a factor of 0.006 (the angular separation in radians). Also, as its offset is predominantly in the east-west direction, it samples similar source zenith angles as the target masers, which further reduces the effects of unmodeled atmospheric delays [...] We used a strong H2O maser spot as the interferometer phase-reference, because it was considerably stronger than the background source and could be detected on individual baselines in the available on-source time as short as 8 s."[8]

The image at right is from the Very Large Array (VLA) "90-cm wavelength image, adapted from LaRosa et al. (2000), with the locations of Sgr A*, Sgr B2, and J1745–2820 indicated."[8]

## Objects

Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to celestial objects including to the Moon/Keynote lecture, the Sun, and to stars beyond the Solar System.

## Electromagnetics

"The majority of rotation measurements (RMs) have been made to pulsars in quadrant 1. An analysis of RMs by Rand & Lyne (1994) has the uniform component of the local magnetic field directed towards ℓ ~ 90° with a magnitude of ~ 1.4 μG. About 400 pc towards ℓ = 0° the field direction reverses."[9]

## Continua

In order to obtain parallax measurements, a number of 'fixed' sources must be available to determine the parallax motion. For a parallax measurement of Cygnus X-1 it is necessary to have "the positions of Cygnus X-1 and the background continuum sources, as well as their angular separations from Cygnus X-1."[10]

At the right is an image of the two continuum 'fixed' radio sources J1953+3537 at J2000.0 R.A. 19h 53m 30.875712s Dec. +35° 37' 59.35927" and J1957+3338 at J2000.0 R.A. 19h 57m 40.549923s and Dec. +33° 38' 27.94339".[10] These are needed to measure the parallax movement of Cygnus X-1.

## Emissions

Pulsars "PSRs J1744−1134 and J1024−0719 are two of only three isolated [millisecond pulsars] MSPs detected at X-ray energies. Since these MSPs have similar spin parameters [...], and presumably similar evolutionary histories, comparing their X-ray properties is useful in understanding the origin of their X-ray emission. With our revised distance estimates, the X-ray luminosity of PSR J1024−0719, Lx < 1 × 1029d2erg s−1, is less than a third of the previously accepted value, and for PSR J1744−1134, Lx = 4 × 1029d2erg s−1, is slightly higher than the previous value, and much larger than that of PSR J1024−0719".[9]

## Absorptions

"When detection of neutral hydrogen (HI) absorption of the pulsar signal is possible, an estimate, or at least a limit on the distance may be obtained using a Galactic rotation model".[9]

"There is strong evidence for an elongated cavity in the neutral component of the [local insterstellar medium] LISM. This cavity surrounds the Sun and extends several hundred parsecs into quadrant 3 (Lucke 1978). The cavity appears as a region of low reddening extending 500 pc between ℓ = 210° and 255° and 1.5 kpc toward ℓ = 240°. Running counter to this is very heavy obscuration beyond ~100 pc in the first quadrant. Similarly, HI column densities derived from ultraviolet observations show a marked paucity in HI along LOSs directed towards ℓ = 230° (Frisch & York 1983; Paresce 1984). A similar morphology for this cavity is gleaned from NaI absorption measurements".[9]

## Bands

"The timing measurements of PSR J1744−1134 were made as part of an ongoing millisecond pulsar (MSP) timing project using the 64-m Parkes radio telescope. [...] Between 1995 January and 1999 January we made regular observations of PSR J1744−1134 at 0.66 and 1.4 GHz. At 0.66 GHz we used a dual linear polarization receiver. During the period of 1997 April until 1998 August we used the center beam of the Parkes Multibeam receiver system for dual linear polarization observations about 1.4 GHz. At other times observations at 1.4 GHz were made with a dual circular polarization H-OH receiver. The downconverted signal was fed into the Caltech correlator (Navarro 1994) where it was digitized and autocorrelated. The autocorrelation functions were folded at the topocentric pulse period, Fourier transformed, and compressed to 180 s sub-integrations, 8 frequency channels and 512 phase bins. A typical observation consisted of 8 contiguous sub-integrations. At 0.66 GHz, the signal was recorded over 32 MHz of bandwidth. Near 1.4 GHz we observed with two 128 MHz bands centered near 1.4 and 1.6 GHz respectively."[9]

## Backgrounds

The "NRAO Very Large Array (VLA; program AR677) [has been used] in the most extended A configuration on 2008 September 28 to find background extragalactic sources as position references near the target H2O maser sources."[11]

We "selected unresolved sources in the NRAO VLA Sky Survey (NVSS) catalog within ≈ 2°of each maser target and observed both background candidates and target masers at 8 and 22 GHz. For W51 Main/South, we found one new background source J1922+1504 for VLBA observations in addition to two known calibrators from the VLBA Calibrator Survey-1 (VCS1; Beasley et al. 2002), J1922+1530 and J1924+1540. However, at 0.3 mas resolution of the VLBA, we found the new source J1922+1504 displayed a resolved structure and was not useful for parallax measurements"[11]

"Using a four-block setup allows the middle block of observing time to be available for phase-referencing rapid-switching scans between the target maser and background continuum sources when the source elevation was the highest at most stations. We measured multi-band delays and fringe rates, mostly due to un-modeled atmospheric propagation delays, to determine zenith delay errors as a function of time for each antenna."[11]

## Meteors

"A PDM operates in the following four trigger modes: a) Normal mode with a GTU of 2.5 μs for routine data taking of EAS, b) Slow mode with a programmable GTU up to a few ms, for the study of meteorites and other atmospheric luminous phenomena, c) Detector calibration mode with a GTU value suitable for the calibration runs, and d) Lidar mode with a GTU of 200ns."[12]

"The Laser of the [Japanese Experiment Module (JEM) of ISS for the Extreme Universe Space Observatory] JEM-EUSO Lidar releases the short pulse (less than 10ns, 20mJ/pulse) of the UV photons with 355nm in the frequency of 50Hz. The returned pulses are observed by the main telescope in a higher time resolution (200ns) of the Lidar mode of the [Photo-Detector Module] PDM. The slow data of the main telescope can also be used to determine the cloud top height by trigonometric parallax."[12]

## Cosmic rays

"Evidences of non-thermal X-ray emission and TeV gamma-rays from the supernova remnants (SNRs) has strengthened the hypothesis that primary Galactic cosmic-ray electrons are accelerated in SNRs."[13]

"So far, the canonical distance to the Vela SNR has been taken to be 500pc, a value which was derived from the analysis of its angular diameter in comparison with the Cygnus Loop and IC443 (Milne 1968), and pulsar dispersion determination (Taylor & Cordes 1993). However, recent parallax measurements clearly indicate that the distance of 500pc is too large. Cha et al. (1999) obtained high resolution Ca-II absorption line toward 68 OB stars in the direction of the Vela SNR. The distances to these stars were determined by trigonometric parallax measurements with the Hipparcos satellite and spectroscopic parallaxes based upon photometric colors and spectral types. The distance to the Vela SNR is constrained to be 250 ± 30pc due to the presence of the Doppler spread Ca-II absorption line attributable to the remnant along some lines of sight. Caraveo et al. (2001) also applied high-resolution astrometry to the Vela pulsar (PSR B0833-45) V ∼ 23.6 optical counterpart. Using Hubble Space Telescope observations, they obtained the first optical measurement of the annual parallax of the Vela pulsar, yielding a distance of 294+76 −50 pc. Therefore, we calculate the electron flux adopting a distance of 300 pc to the Vela SNR."[13]

## Neutrals

"To further characterize the distribution of electrons in the LISM it is useful to relate their location to other interstellar features, such as bubbles, superbubbles, and clouds of neutral gas. There is strong evidence for an elongated cavity in the neutral component of the LISM. [...] There are several features of interest within this cavity. One of these is the local hot bubble (LHB): a volume encompassing the Sun distinguished by low neutral gas densities and a 106 K, soft X-ray emitting gas"[9]

The "neutral hydrogen column density [has] a level of N(HI)= 5 × 1019 cm−2"[9]

## Neutrons

Abundance "estimates for neutron-capture elements, including lead (Pb), and nucleosynthesis models for their origin, in [the] carbon-rich, very metal-poor [star], [...] LP 706-7 [are reported]. [...] A Pb abundance is also derived for LP 706-7 by a re-analysis of a previously observed spectrum."[14]

"LP 706-7 [was] observed with the University College London coudé échelle spectrograph (UCLES) and Tektronix 1024×1024 CCD at the Anglo-Australian Telescope. [...] the numbers of photons obtained around the Pb I λ4057 are [...] 3000 per 0.04Å pixel (S/N∼80) for [...] LP 706-7".[14]

"The surface gravity of LP 706-7 (Norris et al. 1997a) was based on the requirement that Fe I and Fe II lines give identical abundances. More recently, a trigonometric parallax for this star has been published from the Hipparcos mission (ESA 1997), π = 15.15 ± 3.24 mas. Somewhat surprisingly, this surface gravity indicates an absolute magnitude MV = 8.0 ± 0.4, which is subluminous compared to both main sequence and subgiant Population II stars with Teff = 6000 K. A subgiant of MV = 3.0 or 4.0 would have a parallax of only 1.5 or 2.4 mas. Either the Hipparcos measurement of this star is significantly in error, or the star is far more bizarre than its CH-star status suggests. If the temperature estimate (based on photometric colors) and the Hipparcos parallax were both correct, we should be forced to infer a radius ten times smaller than for a subgiant and four times smaller than for a main-sequence star, but the surface gravity appears inconsistent with such a compact object (since g ∝ M/R2). It seems most likely that the Hipparcos parallax is simply incorrect, although an examination of the records (D. W. Evans, priv. comm.) revealed no concerns."[14]

For "LP 706-7, because radial-velocity variations that might be expected for a star with a white-dwarf companion have not yet been detected (Norris et al. 1997a)."[14]

We "found strong excesses of neutron-capture elements in the two metal-deficient satrs LP 625-44 and LP 706-7 with [Fe/H]= −2.7 and −2.74, respectively, which are interpreted as the result of s-process nucleosynthesis from a single site. Namely, the abundant material polluted by s-process nucleosynthesis dominates over the original surface abundances of neutron-capture elements. For instance, the Ba abundance in these two stars is a factor of several hundred times higher than the general trend of model predictions at [Fe/H]= −2.7. Even the abundance of Eu, which is usually interpreted as a signature of the r process, but should also be produced by the s-process as well, is enhanced by more than a factor of 10 in these two stars. Therefore, the neutron-capture elements in these two stars should present almost pure products of s-process nucleosynthesis at low metallicity. The exceptions to this are the abundances of Sr and Y in LP 706-7, which show no distinct excess. Therefore, the contribution of the s-process to these two elements may not be significant for this star."[14]

There "is no evidence of binarity for [...] LP 706-7 (Norris et al. 1997a)."[14]

The "precise mechanism for chemical mixing of protons from the hydrogen-rich envelope into the 12C-rich layer is still unknown, even for stars with solar metallicity, despite several theoretical efforts (Herwig et al. 1997; Langer et al. 1999). This makes it even harder to understand the peculiar abundance pattern of the s-process elements found in carbon-rich, metal deficient stars such as LP 625-44 and LP 706-7."[14]

What "physical conditions are necessary to reproduce the observed s-process abundance profile of LP 625-44 and LP 706-7 without adopting any specific stellar model."[14]

As "long as the same neutron exposure is adopted, the abundance patterns of LP 625-44 and LP 706-7 are reproduced with equivalent reduced χ2 values, even in extreme conditions of very high neutron density, Nn ≳ 1011cm−3. These parameter values simulate, more or less, the s-process conditions expected during the thermal pulse phase (Iben 1977)."[14]

"Almost all elements, except for Pb, were found to be made in the first neutron exposure. Even the lead abundance converges after about three recurrent neutron exposures. This is consistent with the small overlap factor, r ≈ 0.1, deduced in our best-fit model. [...] fixed neutron exposure τ = 0.71 for LP 625-44. The observed Pb/Ba ratio is reproduced in the few-pulse model only for a small overlap factor, r ≲ 0.2, while the Ba/Sr ratio is rather insensitive to r and allows for a wider range, r ≲ 0.65. The Pb abundance is so sensitive to r that large r-values (0.2 ≲ r) are almost entirely excluded [...] This is a characteristic feature of the s-process pattern observed in LP 625-44 and LP 706-7."[14]

"The ratio is slightly higher in LP 706-7, [Pb/Ba] = +0.27 ± 0.24. This may indicate that a range of 13C amounts is indeed required in the most metal-poor AGB stars, as well as for the moderately metal-poor ones."[14]

## Protons

"The Hummer & Mihalas theory is used to describe the non-ideal effects due to perturbations on the absorber from protons and electrons. We use a truncation of the electric microfield distribution in the quasi-static proton broadening to take into account the fact that high electric microfields dissociate the upper state of a transition."[15]

"For the lower Lyman lines (Lα, Lβ, and Lγ), close range collisions of the absorber with hydrogen atoms and protons cause the appearance of important satellites in the wings of the lines that are visible up to Teff ∼ 30, 000 K".[15]

"The Stark effect is defined as the shifting — or splitting — of spectral lines under the action of an electric field. [In] the atmospheric plasma a local electric microfield [is] due to protons that is constant with time."[15]

"Classically, the sum of the electric potentials of the absorber and the nearby protons only allows bound states in the local potential minima up to a certain energy, called the saddle point."[15]

"The most reliable independent observational constraint for DA white dwarfs comes from trigonometric parallax measurements. [...] there exists a very good correlation between spectroscopically based photometric distance estimates and those derived from trigonometric parallaxes. Here we compare absolute visual magnitudes instead of distances. We first combine trigonometric parallax measurements with V magnitudes to derive MV (π) values. We then use the calibration of Holberg & Bergeron (2006) to obtain MV (spec) from spectroscopic measurements of Teff and log g. [...] the bright white dwarf 40 Eri B for which a very precise trigonometric parallax and visual magnitude have been measured by Hipparcos. These measurements yield MV = 11.01 ± 0.01, [...] atmospheric parameters determined from the spectroscopic technique remain in excellent agreement with the constraints imposed by trigonometric parallax measurements."[15]

## Electrons

"The parallax distance of 357+43 −35 pc is over twice that derived from the dispersion measure using the Taylor & Cordes model for the Galactic electron distribution."[9]

"The mean electron density in the path to the pulsar, ne = (8.8±0.9) × 10 −3 cm−3, is the lowest for any disk pulsar."[9]

Comparing "the ne for PSR J1744−1134 with those for another 11 nearby pulsars with independent distance estimates[, ...] there is a striking asymmetry in the distribution of electrons in the local interstellar medium. The electron column densities for pulsars in the third Galactic quadrant are found to be systematically higher than for those in the first. The former correlate with the position of the well known local HI cavity in quadrant three. The excess electrons within the cavity may be in the form of HII clouds marking a region of interaction between the local hot bubble and a nearby superbubble."[9]

The "pulsar distances provide information about the column density of free electrons along different lines-of-sight (LOSs). Since their discovery, pulsars have had their distances estimated primarily by measuring the dispersion delay between pulses arriving at two widely spaced frequencies. As this delay is a function of the integral of electron density along the LOS to the pulsar, or dispersion measure (DM), a model for the Galactic free electron distribution yields the pulsar’s distance".[9]

A "measurement of the parallax of PSR J1744−1134 obtained by analysing pulse times of arrival (TOAs) spanning 4 years [is combined] with other similar measurements to study the electron distribution in the local ISM (LISM)."[9]

Parallax "measurement of PSR J1744−1134 places it at a distance of 357+43 −35 pc. This distance is over twice the value of 166 pc derived from the DM using the Taylor & Cordes (1993) model. The improved distance measurement implies a mean electron density in the path to the pulsar of ne = (8.8±0.9) × 10−3 cm−3."[9]

"When the distances to the sample of 12 local pulsars are projected onto the Galactic plane a striking asymmetry in the electron distribution becomes evident. [...] the LOS electron densities to pulsars in the third Galactic quadrant are systematically higher than those in the first. [...] The asymmetry may be more pronounced since the upper parallax limit of PSR B1929+10 sets an upper limit to its electron density (ne < 0.013 cm−3), while the Shklovskii effect implies a firm lower bound to the ne of PSR J1024−0719 (ne > 0.029 cm−3)."[9]

The "large number of pulsars with lower than expected electron densities in quadrant 1 implies a dearth of ionized material in this quadrant at least out to ~ 1 kpc."[9]

"In the case of the prompt release of electrons after the explosion, the flux from the Vela SNR is the largest among the known SNRs [...] The flux value is quite sensitive to the change of distance to Vela from 500pc to 300pc, since the solution for electron density yields a Gaussian distribution function of r [...]. The flux of electrons at a distance of 300pc is two orders of magnitude larger than at 500pc. [...] the Vela SNR is the most dominant source in the TeV region."[13]

## Positrons

"An important example is the studies of the magneto-ionic medium in the Milky Way using the Faraday rotation measure RM and dispersion measure DM of pulsars – both are integrals along the line of sight involving interstellar magnetic field and thermal electron density."

"A pulsar’s own contribution to the observed RM is minor because the pulsar magnetosphere is populated by electron-positron pairs resulting in zero net Faraday rotation."

"Distance estimates now exist for a few hundreds of pulsars, resulting from three basic techniques: neutral hydrogen absorption (in combination with the Galactic rotation curve), trigonometric parallax and from associations with objects of known distance".[16]

## Muons

"The issue of autonomously detecting satellite and airplane tracks in images is by no means a new one. For decades, these tracks have been nothing more than a nuisance for astronomers–foreground artifacts that must be disposed of in the preprocessing of data [...] the Recognition by Adaptive Subdivision of Transformation Space algorithm [1] removes satellite streaks directly from images using a geometric approach that assumes the tracks are straight lines [...] the Random Sampling and Consensus algorithm [allows] for postprocessing removal of curved tracks and scratches as well."[17]

"While these streaks may be a source of noise in the field of astronomy, for applications such as the Space Surveillance Network (SSN), they are the signal. A track from a satellite or piece of debris, along with time-stamp information, allows the SSN to make an equatorial angles-only determination of its orbit. [A way of] obtaining the timestamp information [...] is to measure the start and end times of an exposure and extract the end points of the imaged track(s). [...] [For determining the range of an artificial satellite using its observed trigonometric parallax] the error in detecting the end points may very well dominate the other sources of error in the measurement [3]."[17]

The purpose of the "Space-based Telescopes for Actionable Refinement of Ephemeris (STARE) [...] is to refine orbital information for satellites and debris by directly imaging them with CMOS imagers onboard a constellation of cube-satellites (CubeSats). The images acquired by a given sensor will be run through an algorithm in the onboard microprocessor that is tasked with extracting star and track end point coordinates and sending them to the ground (without the accompanying image). Since the attitude of the STARE satellites will not be precisely controlled, the telescopes may be rotating about the pointing axis."[17]

"Once a contiguous set of pixels has been identified, it is characterized as a star, track, or unknown object (such as a delta or Compton scattered worm) based upon its ellipticity (e) and the number of pixels (N) it contains. These values are dependent on the optical system and detector used, but for STARE, a cut of e > 0.8 and N > 20 should effectively identify all real tracks. A perfectly straight track should have e = 1; the margin e = 0.8–1.0 allows for curvature and the possibility of overlapping stars or cosmic rays. The chance of a muon hit producing a track greater than 20 pixels long is extremely low."[17]

## Neutrinos

Hot "coronae disappear slowly in the course of stellar evolution, since the evolutionary tracks are parallel to the revised X-ray dividing line and do not cross it. Furthermore, hybrid stars turn out to be quite common: they are just more massive, more active and maintain some coronal plasma beyond the onset of cool [stellar] winds."[18]

"For the nearest (d < 18pc), MV is derived from the trigonometric parallax, and interstellar extinction can be neglected. For larger and trigonometrically less reliable distances, [use] the Wilson-Bappu magnitudes [...] For most of the more distant hybrid stars, extinction is non-negligible."[18]

Evolutionary tracks [have been] updated "for the most recent opacities (i.e., OPAL), nuclear rates, neutrino losses and a refined equation of state".[18]

## Gamma rays

"The launch of the Fermi Gamma-ray Space Observatory in June 2008 completely changed the status in studies of gamma-ray pulsars. The first published catalog of gamma-ray pulsars (Abdo et al. 2010) contains 46 gamma-ray pulsars including 8 millisecond pulsars, 21 young radio pulsars and 17 gamma-selected pulsars. After more than one and half years of all-sky survey observations by Fermi/LAT, more than 70 gamma-ray pulsars were discovered, including 25 gamma-selected pulsars (see reviews by Ray & Saz Parkinson 2010). High sensitivity of the Fermi/LAT makes a new era for pulsar discoveries, specially for the population of radio-quiet gamma-ray pulsars."[19]

"Trigonometric parallax measurements of radio pulsars are the reliable method, but are only available for the nearby pulsars (< 0.4 kpc) specially for a few radio millisecond pulsars [...] For the Geminga pulsar, we estimate the distance of 0.19 ± 0.07 kpc which is well consistent with the distance value of 0.25+0.12 −0.06 kpc from the optical trigonometric parallax measurement (Faherty et al. 2007). [...] [For m]illisecond pulsars (MSPs) [...] their distances are generally measured by optical trigonometric parallax".[19]

## X-rays

Trigonometric parallax may be somewhat wavelength dependent. Usually, it depends on resolution and using a periodic set of measurements where the diameter of the period is large enough to allow resolution. To attempt X-ray trigonometric parallax, a candidate star that is within resolution of at least one currently available or formerly available X-ray satellite is needed. If none qualify for even the closest star, then a calculation that is time based may work or a calculation using a greater periodic set is needed.

For the effort to succeed a collection of three to five very distant, unmoving X-ray sources must be available to determine the target's relative movement.

Since the idea of using X-ray astronomy satellites is novel and perhaps a bit premature, finding a relatively nearby source where the experiment may be performed may be itself a time-consuming task too soon for a course.

According to SIMBAD, V645 Cen (Proxima Centauri) is an X-ray source in the catalogs: 1E, 2E, 1ES, RE, RX, 1RXS, and [FS2003]. Catalogs 1E through 1ES are the Einstein satellite observations. Catalogs RE through 1RXS are the ROSAT satellite observations. Catalog [FS2003] is a systematic search for variability among ROSAT All-Sky Survey X-ray sources by B. Fuhrmeister and J. H. M. M. Schmitt in an article published in Astronomy and Astrophysics.

The X-ray resolutions of these two satellites are

1. High Energy Astrophysics Observatory 2 (Einstein X-ray Observatory) - "a spatial resolution of ~1´."[20] and
2. ROSAT - "~ 2 arcsec spatial resolution (FWHM)".[21]

The visual astronomy parallaxes (mas) on Proxima Centauri are 774.25 ± 2.08, according to SIMBAD. As the resolution of neither Earth-orbit satellite is within the parallax range it is unlikely that any sets of X-ray observations from these two satellites can resolve parallax movement by Proxima Centauri.

Perhaps one of the current X-ray satellites has sufficient resolution:

1. AGILE - not stated so far,
2. Chandra X-ray Observatory (Advanced X-ray Astrophysics Facility, (AXAF)) - Spatial resolution < 1 arcsec, HRC-I ~ 0.5 arcsec spatial resolution,
3. Fermi Gamma-ray Space Telescope - not stated so far,
4. INTEGRAL - spatial resolution 3´,
5. MAXI - not stated so far,
6. NuSTAR (Nuclear Spectroscopic Telescope Array Mission) - not stated so far,
7. Suzaku (Astro-E2) - angular resolution of ~2´ (HPD); all gold-coated,
8. Swift X-ray Telescope (XRT) - ~5 arcsec position accuracy, and
9. X-ray Multi-Mirror Mission (XMM-Newton) - Spatial resolution 6" FWHM. None of these can perform trigonometric parallax at present, although Chandra may be able to estimate the parallax.

"Chandra and XMM-Newton observations of the red dwarf star Proxima Centauri have shown that its surface is in a state of turmoil. Flares, or explosive outbursts, occur almost continually. This behavior can be traced to Proxima Centauri's low mass, about a tenth that of the Sun. In the cores of low mass stars, nuclear fusion reactions that convert hydrogen to helium proceed very slowly, and create a turbulent, convective motion throughout their interiors. This motion stores up magnetic energy which is often released explosively in the star's upper atmosphere where it produces flares in X-rays and other forms of light."[22]

"The same process produces X-rays on the Sun, but the magnetic energy is released in a less explosive manner through heating loops of gas, with occasional flares. The difference is due to the size of the convection zone, which in a more massive star such as the Sun, is smaller and closer to its surface."[22]

"Red dwarfs are the most common type of star. They have masses between about 8% and 50% of the mass of the Sun. Though they are much dimmer than the Sun, they will shine for much longer - trillions of years in the case of Proxima Centauri, compared to the estimated 10 billion-year lifetime of the Sun."[22]

"X-rays from Proxima Centauri are consistent with a point-like source. The extended X-ray glow is an instrumental effect. The nature of the two dots above the image is unknown - they could be background sources."[22]

"Image is 1.5 arcmin across."[22]

## Ultraviolets

"The M8 brown dwarf 2MASSW J1207334-393254 (hereafter 2M1207A) is [...] a [...] brown dwarf [member] of the ~ 10 Myr old TW Hydrae Assocation (Webb et al. 1999). 2M1207A is a very-low-mass substellar analog to a classical T Tauri star: It has broad, variable Hα emission due to accretion [...], mid-infrared excess due to a disk [...], ultraviolet emission due to hot accreted gas and warm circumstellar molecular hydrogen gas [...], and forbidden oxygen emission due to an outflow[, but] it is not detected in X rays [...] or radio [...], so is apparently relatively magnetically inactive."[23]

A "red companion (2M1207B), [is] 5 magnitudes fainter in the K band. Common proper motion confirms that this is a bound pair [...] with a separation of 773 ± 1.4mas. The secondary has a late-L spectral type (Mohanty et al. 2007). The inferred luminosity implies a mass ~ 5MJ [...], although Mohanty et al. (2007) suggest that the secondary is 8 ± 2 jupiter masses and viewed through an edge-on disk."[23]

"Because the TWA is a relatively nearby, loose association there has been some confusion on the distance to the system. Chauvin et al. (2004) adopted a distance of 70 pc, on the basis of theoretical models of brown dwarf evolution. The Hipparcos distance of TW Hya itself is 56+8 −6pc (Perryman et al. 1997). Mamajek (2005) used the moving cluster distance method to estimate the distance to 2M1207A to be 53 ± 7 pc, while Song et al. (2006) used the same method, but an updated proper motion and a different group membership list to estimate 59 ± 7 pc. With uncertainties in the distance to the TW Hya group of ~ 15%, firm conclusions about the natures of 2M1207 A and B, as well as other members of the group, have been elusive.” Here we present the first trigonometric parallax for 2M1207A. We confirm that it is a member of the TW Hya Association and put ... constraints on the planet candidate 2M1207B."[23]

"Observations of 2M1207A in the IKC band were obtained at the CTIO 0.9m telescope by the RECONS group via the SMARTS Consortium. There are 54 parallax frames obtained over 2.14 years. [...] The resulting relative parallax is πrel = 17.93 ± 1.03 mas. VRI photometry was obtained in July 2007 on five nights using the same telescope [...] We estimate the correction to absolute parallax to be 0.58 ± 0.05 mas on the basis of photometry of the seven reference stars [...] The absolute parallax is therefore 18.51 ± 1.03 mas, for a distance of 54.0+3.2 −2.8pc. The observed proper motion is 66.7 ± 1.5 masyr−1 at position angle θ = 250.0 ± 2.4 degrees."[23]

"The distance and proper motion of 2M1207 is consistent with TWA membership. The position angle expected for motion towards [the] TWA convergent point is 251.4 degrees, consistent with the measured proper motion. Using [a] radial velocity of +11.2±2.0 km s−1 for 2M1207A, the (U,V,W) space velocities are (−8,−18,−4) kms−1, consistent with [the] centroid group value of (−10.2,−17.1,−5.1) kms−1. In particular, the measured distance rules out any association with the background Lower Centaurus Crux [other] measurements and our distance, the projected separation is 41.7 ± 2.3 A.U."[23]

"2M1207A to be 24±6MJ brown dwarf. Because they used [the] value of 53 pc as the distance, this mass is not changed significantly by a distance increase of 2%: 2M1207A is best understood as a ~ 25MJ brown dwarf. The disk parameters [...] also remain unchanged because they used the same distance. The observed V − Ks = 8.00 ± 0.19 is consistent with the M8 spectral type and suggests the accretion rate at the time was ≲ 10−11Myr−1 [...] Like the young M dwarf AU Mic (Gl 803), 2M1207A lies ~ 1.5 magnitudes above the main sequence in the MV vs V − K diagram".[23]

"The usual procedure for analyzing 2M1207B is to assume a bolometric correction appropriate to late-L dwarfs, and then fit the luminosity to evolutionary models. [Estimates are] 5 ± 2MJ for 70 pc, [...] 5 ± 3MJ for 59 pc, and [...] 3 − 4MJ for 53 pc. The trigonometric parallax would therefore support the last two estimates. [...] H and K-band near-infrared spectra of 2M1207B were best fit by an effective temperature of 1600 ± 100K. However, for the 3 − 5MJ fits, the expected effective temperature is more like 1000 − 1200 K. [Perhaps] the best resolution is that 2M1207B is viewed through an edge-on gray disk, and that therefore it is more luminous than otherwise estimated. 2M1207B is then a 8 ± 2MJ planetary mass brown dwarf. The wide separation and mass ratio (q ≈ 0.2 − 0.3) suggests this planetary-mass object did not form through core accretion [...For] 2M1207B [...] it is red compared to field brown dwarfs, which can be attributed to having more dust in the photosphere."[23]

## Opticals

"From the accurate trigonometric parallax [...], the effective temperature (Teff = 10, 900 K) and the stellar radius (R = 0.00368 R) are directly determined from the broad-band spectral energy distribution — the parallax method. The effective temperature and surface gravity are also estimated independently from the simultaneous fitting of the observed Balmer line profiles with those predicted from pure-hydrogen model atmospheres— the spectroscopic method (Teff = 10, 760 K, log g = 9.46). The mass of LHS 4033 is then inferred from theoretical mass-radius relations appropriate for white dwarfs. The parallax method yields a mass estimate of 1.310–1.330M, for interior compositions ranging from pure magnesium to pure carbon, respectively, while the spectroscopic method yields an estimate of 1.318–1.335 M for the same core compositions. This star is the most massive white dwarf for which a robust comparison of the two techniques has been made."[24]

"LHS 4033 (WD 2349−031) is a white dwarf [that] has also been part of the Luyten Half Second (LHS) survey μ ≥ 0.6′′ yr−1 white dwarf sample [...] virtually all of which have been targeted for accurate trigonometric parallaxes at the U.S. Naval Observatory, for purposes of estimating the luminosity function of cool white dwarfs."[24]

Here are the "optical and infrared photometry for LHS 4033":[24]

1. V = 16.98 ± 0.02
2. B–V = +0.19 ± 0.03
3. V – I = +0.07 ± 0.03
4. J = 16.97 ± 0.05
5. J–H = +0.05 ± 0.07
6. H–K = −0.10 ± 0.07
7. πabs (mas) = 33.9 ± 0.6
8. μrel (mas yr−1) = 701.4 ± 0.2

PA (deg) = 66.3 ± 0.1 Distance (pc) = 29.5 ± 0.5 MV = 14.63 ± 0.04

"Since the spectral type of LHS 4033 is DA and non-magnetic, the mass may be estimated by fits to the Balmer lines (see, e.g., Bergeron et al. 1992) in a much more rigorous fashion. The surface gravity used with suitable evolutionary models yields independent determinations of the mass and radius. The effective temperature may also be estimated from broad-band photometry once the dominant atmospheric constituent is known. This, along with an accurate trigonometric parallax, permits a different estimate of the luminosity, radius, and mass (Bergeron et al. 2001). While it has been possible to compare the parameter determinations of these methods for limited samples of white dwarfs, it is particularly interesting to do so for a massive star."[24]

"Trigonometric parallax observations were carried out over a 6.05 year interval (1997.76 – 2003.81) using the USNO 1.55 m Strand Astrometric Reflector equipped with a Tek2K CCD camera (Dahn 1997). The absolute trigonometric parallax and the relative proper motion and position angle derived from the 150 acceptable frames [...]. The parallax and apparent V magnitude then yield an absolute magnitude [...]."[24]

Optical "spectroscopy was secured on 2003 October 1 using the Steward Observatory 2.3-m reflector telescope equipped with the Boller & Chivens spectrograph and a UV-flooded Texas Instrument CCD detector. The 4.5 arcsec slit together with the 600 lines mm−1 grating blazed at 3568 Å in first order provided a spectral coverage of 3120–5330 Å at an intermediate resolution of ~ 6 Å FWHM. The 3000 s integration yielded a signal-to-noise ratio around 55 in the continuum."[24]

"We first assume log g = 8.0 and determine the effective temperature and the solid angle, which, combined with the distance D obtained from the trigonometric parallax measurement, yields directly the radius of the star R. The latter is then converted into mass using an appropriate mass-radius relation for white dwarf stars. Here we first make use of the mass-radius relation of Hamada & Salpeter (1961) for carbon-core configurations. This relation is preferred to the evolutionary models of Wood (1995) or those of Fontaine et al. (2001), which extend only up to 1.2 and 1.3M, respectively. [...] In general, the value of log g obtained from the inferred mass and radius (g = GM/R2) will be different from our initial assumption of log g = 8.0, and the fitting procedure is thus repeated until an internal consistency in log g is achieved. The parameter uncertainties are obtained by propagating the error of the photometric and trigonometric parallax measurements into the fitting procedure."[24]

Our "spectroscopic solution Teff = 10, 760 ± 150 K and log g = 9.46 ± 0.04, which translates into M = 1.335± 0.011 and R = 0.00358± 0.00019 R using the Hamada-Salpeter mass-radius relation for carbon-core configurations, is in excellent agreement with the solution obtained with the photometry and trigonometric parallax method. This is arguably the most massive white dwarf subjected to a rigorous mass determination [...]. Note that despite the extreme surface gravity of LHS 4033, the Hummer-Mihalas formalism used in the line profile calculations remains perfectly valid, since the density at the photosphere remains low (ρ ~ 10−5 g cm−3) as a result of the high opacity of hydrogen at these temperatures."[24]

The "parallax method with the Mg configurations yields a mass of 1.310 M (instead of 1.330 when C configurations are used), while the spectroscopic method yields a mass of 1.318 M (instead of 1.335 M)."[24]

## Visuals

For "the prototype variable RR Lyr, [...] the parallax inferred [...] appears in close agreement with Hubble Space Telescope absolute parallax."[25]

We "assumed an intrinsic luminosity of RR Lyr in the range log L/L= 1.65–1.80 to cover current uncertainties of HB models. On this basis, we derived MK=−0.541 ± 0.062 mag and a ‘pulsation’ parallax πpuls= 3.858 ± 0.131 mas, in close agreement with the HST absolute value πabs= 3.82 ± 0.20 mas."[25]

Avoiding "any assumption about the RR Lyr bolometric luminosity. Adopting log P=−0.2466 [...], K= 6.54 mag [...], [Fe/H]=−1.39 ± 0.15 [...] and V= 7.784 mag [...] [using]

${\displaystyle (M_{V}-M_{K})^{FO}=5.180+2.518\cdot log(P)-0.168\cdot log(Z)-2.158\cdot log(L)}$

gives log L= 1.642 ± 0.024 + 0.535AV. [For] each adopted extinction correction, one derives directly from the observed V−K colour the luminosity value to be inserted into"[25]

${\displaystyle M_{K}^{F}=0.565-2.101\cdot logP+0.125\cdot logZ-0.734\cdot logL,}$

"without using any evolutionary predictions. Once MK is determined from [the above equation], both the intrinsic distance modulus (μ0=K− 0.11AVMK) and the absolute visual magnitude (MV=VAV−μ0) can be determined easily."[25]

Despite "the substantial improvement in the accuracy of the RR Lyr trigonometric parallax provided by HST, when compared with previous measurements [...] a sound empirical determination of its absolute magnitude is still hampered by the intrinsic uncertainty in the HST measurement, even if the interstellar extinction to RR Lyr was known firmly."[25]

Both "V and K absolute magnitudes of RR Lyr itself, estimated via the pulsational approach, are in good agreement with the trigonometric parallax recently measured by HST [...]. This suggests that at least for the metallicity of RR Lyr ([Fe/H]=−1.39), the pulsational approach is consistent with direct distance determinations."[25]

## Violets

"Hipparcos trigonometric parallaxes and photometric data [are available] for about 40 bright carbon stars [...] Individual absolute visual and bolometric magnitudes, normal color indices [blue (B), violet (V)] (BV)0, absorption values and distance moduli were determined. By comparison with stellar evolutionary tracks for initial mass 1 ≤ M/M ≤ 4 it is found that the majority of CH- and R-stars are on the giant and subgiant branches, but N-stars occupy a region −4 < MV < −1 and 1.6 < (BV)0 < 3.6 and correspond to an advanced stage of thermally pulsing asymptotic branch giants."[26]

"Using Hipparcos parallaxes and proper motions, three multiple stars with a carbon star component are examined. Hipparcos data confirms a physical link between W CMa and HD 54306 (B2V), both probable members of the association CMa OB1. Some stars are located below the subgiant branch for the mass 1M and a number of the N stars are below the theoretical limit for carbon stars on the AGB."[26]

"The most straightforward method, i.e. through trigonometric parallaxes, has hitherto been of little value, owing to the considerable distances even to the nearest carbon stars, and the imperfectness of previous measuring methods [...] The situation has radically changed after the mission by the astrometric satellite Hipparcos. The mean error of about 1 mas – a characteristic value for parallaxes measured by Hipparcos – provides us with reliable distance estimates inside, say, the 0.5 kpc region around the Sun including some 100 carbon stars."[26]

"Hipparcos also supplied us with precise photometric data, giving the mean brightness estimate from ~ 100 observations of each star, a circumstance, which because of the variability of carbon stars, is of special value. Here, a problem specific to carbon stars – stars with a peculiar spectral energy distribution –, to accurately correct ground-based photometry for atmospheric extinction, is irrelevant for Hipparcos data."[26]

"Hipparcos results clearly confirm that the great range of observed scatter in the color index BV is intrinsic and not caused by different amounts of interstellar reddening [...] A considerable stretch in the horizontal direction is a result of enhanced sensitivity of the color index (BV)0 to small temperature changes in a cool extended atmospheres; also various degree of violet depression play a definite role."[26]

## Blues

"Observations of [SSSPM J2231-7514 and SSSPM J2231-7515 imaged at right in the violet band] were carried out using the Danish Faint Object Spectrograph (DFOSC) on the Danish 1.54m Telescope in La Silla. Data were taken during the nights starting June 19-20, 2001 (local time) in relatively good almost photometric conditions."[27]

For the blue (B) band, "[A grism was] used, number-7 (3800-6800°A, 5250°A blaze, 1.65°A/pixel resolution – a 1800s and a 1300s spectroscopic observation). Two spectrophotometric standards were taken using [the] grism, LTT 7379 and LTT 9239 [...], as well as a large number of zero, flat (both for imaging and spectroscopy for all grisms used) and arc-lamp (for both grisms) calibration frames. For broad-band photometric calibrations, Landolt standard fields (Landolt 1992) were observed repeatedly [...] a spectrum of one of the objects, SSSPM J2231-7515, which was discovered independently, had already been observed half a year earlier. The spectrum of this star was observed with the EFOSC spectrograph on the ESO 3.6m telescope during the night of 2 December 2000. A slit width of 1.5 arcsec was used with grism number 1 (3185-10940°A, 4500°A blaze, 6.30°A/pixel resolution) for three exposures of 300s each."[27]

A "wide pair (93 arcsec angular separation) of extremely cool (Teff < 4000 K) white dwarfs [have] a very large common proper motion (~1.9 arcsec/yr). The objects were discovered in a high proper motion survey in the poorly investigated southern sky region with δ < −60° using SuperCOSMOS Sky Survey (SSS) data. Both objects, SSSPM J2231-7514 and SSSPM J2231-7515, show featureless optical spectra. Fits of black-body models to the spectra yield effective temperatures of 3810 K and 3600 K, respectively for the bright (V = 16.60) and faint (V = 16.87) component. Both degenerates are much brighter than other recent discoveries of cool white dwarfs with comparable effective temperatures and/or [blue] BJR colours."[27]

"After measuring photometric zeropoints in 6 standard fields (Landolt 1992), we adopted a zeropoint of 24.66 (for 1s exposure time and airmass of 1.45 – the airmass of our acquisition frames) in the Bessel V-band. Examination of the measured zeropoints shows that conditions were very close to photometric (with a hint of very weak cirrus in some cases). Using this zeropoint, we measured the (Vega) magnitudes of the two objects to be 16.60 and 16.87 in V (Bessel). We also measured the magnitude of the extra object, accidentally landing on our 2 arcsec wide slit [...] to be 16.86 [...]. Instrumental magnitude errors are small (1-2%), the main source of error is the determination of the zeropoint. We therefore estimate the overall accuracy of these magnitudes to be better than 5%."[27]

"Spectroscopic frames (science and standard fields and calibration frames) were also bias and zero subtracted, trimmed and flatfielded (using different flat-field frames for the two grisms). The only complication was the removal of focal plane geometric distortions to improve the sky subtraction. This was done by tracing lines in the wavelength calibration frames. A smooth distortion map was fitted to the results and was applied to all frames."[27]

"All objects on the slit were traced along the dispersion axis. Sky subtraction was done through fitting in a 35 arcsec wide band, centered on the object, excluding the central 16 arcsec region. A relatively wide aperture was defined for all objects, which includes all the flux (but degrades the S/N slightly). Object spectra were ‘optimally’ extracted within this aperture, with both cosmic-ray removal (based on photon statistics) and a weighted sum based on estimated signal-to-noise ratio."[27]

"Wavelength calibration was done using He-Ne arc exposures. For the number-7 grism (bluer, higher resolution) the procedure worked quite well (0.06Å RMS, 0.15Å maximum deviation) in the wavelength range 3889-6717Å."[27]

"A detailed photometric and spectroscopic analysis of all known cool white dwarfs (4000 K< Teff < 12000 K) with trigonometric parallax measurements [exists]. [One] cool white [dwarf] with [an] effective [temperature] below 4000 K [...], WD 0346+246, has a trigonometric parallax measurement"[27]

"With the [...] temperatures [...] of 3810 K and 3600 K (3100 K to 3800 K), the newly discovered objects are comparable to the coolest known white dwarf WD0346+246 with 3750 K [...] for which a trigonometric parallax of 36±5 mas (28±4 pc) has been measured".[27]

"If we assume our objects to have the same physical properties (temperature, mass, chemical composition) as WD0346+246, we can estimate their distance from a comparison of their apparent V magnitudes. With V = 19.06 [...], WD0346+246 is more than two magnitudes fainter than our objects (16.60 and 16.87), and consequently we get distance estimates of 9 pc and 10.2 pc. These distance estimates have the same relative uncertainty as for the comparison object, i.e. ±1.3 pc and ±1.5 pc, respectively, and rely on the assumption of identical physical properties, which is unlikely to be the case."[27]

As of 2002, "there are only 11 cool white dwarfs with Teff < 5000 K and trigonometric parallaxes of less than 25 pc [...]. All presently known or suspected degenerates with Teff < 4000 K are at trigonometric or photometric distances of more than 25 pc".[27]

## Infrareds

"Extrasolar-planet searches that target very low-mass stars and brown dwarfs are hampered by intrinsic or instrumental limitations. Time series of astrometric measurements with precisions better than one milli-arcsecond can yield new evidence on the planet occurrence around these objects."[28]

"Over a time-span of two years, we obtained I-band images of the target fields with the FORS2 camera at the Very Large Telescope. Using background stars as references, we monitored the targets’ astrometric trajectories, which allowed us to measure parallax and proper motions, set limits on the presence of planets, and to discover the orbital motions of two binary systems."[28]

"We determined trigonometric parallaxes with an average accuracy of 0.09 mas (≃0.2 %), which resulted in a reference sample for the study of ultracool dwarfs at the M/L transition, whose members are located at distances of 9.5–40 pc."[28]

Astrometric "observations with an accuracy of 120 µas over two years are feasible from the ground and can be used for a planet-search survey."[28]

The image at right is "of DE1520-44 in 0.47" seeing showing the primary (A), its companion (B), and the faint background object (x)."[28]

"The distance [to Cygnus X-1 is] 1.86 +0.12 or -0.11 kpc [...] obtained from a trigonometric parallax measurement using the Very Long Baseline Array."[10]

"Cygnus X-1 was the first [black hole] BH candidate to be established via dynamical observations [...] We observed Cygnus X-1 and two background [continuum] sources over 10 hr tracks at five epochs: 2009 January 23, April 13, July 13, and October 31, and 2010 January 25. These dates well sample the peaks of the sinusoidal trigonometric parallax signature in both right ascension and declination. This sampling provides near maximum sensitivity for parallax detection and ensures that we can separate the secular proper motion (caused by projections of Galactic rotation as well as any peculiar motion of Cygnus X-1 and the Sun) from the sinusoidal parallax effect."[10]

"Generally, data calibration followed similar procedures as for parallax observations of continuum sources in the Orion nebular cluster at 8.4 GHz (Menten et al. 2007). We placed observations of well-known strong sources near the beginning, middle, and end of the observations in order to monitor delay and electronic phase differences among the intermediate frequency bands. In practice, however, we found minimal drifts and used only a single scan of J2005+7752 for this calibration."[10]

The image at the right is a radio signal from Cygnus X-1. It "shows a core–jet structure [...] The peak brightness of Cygnus X-1 ranged between 4 and 9 mJy beam−1 among our observations. [... Including the background continuum sources] restoring beams are in the lower left corner of each panel. All contour levels are integer multiples of 1 mJy beam-1 for Cygnus X-1 and 15 mJy beam-1 for the background sources."[10]

"The change in position of Cygnus X-1 relative to a background continuum source was then modeled by the parallax sinusoid in both coordinates, completely determined by one parameter (the parallax), and a secular proper motion in each coordinate [...] The model included the effects of the ellipticity of Earth's orbit. The weighting of the data in the parallax and proper motion fitting is complicated because the formal position uncertainties are often unrealistically small, since a priori unknown sources of systematic error often dominate over random noise. The north–south components of relative positions often have greater uncertainty than the east–west components because the interferometer beams are generally larger in the north–south direction and systematic errors from unmodeled atmospheric delays usually are more strongly correlated with north–south positions [...] In order to allow for, and estimate the magnitude of, systematic errors, we assigned independent "error floors" to the east–west and north–south position data and added these floors in quadrature with the formal position-fitting uncertainties. Trial parallax and proper motion fits were conducted and a reduced χ2ν (per degree of freedom) statistic was calculated separately for the east–west and north–south residuals. The error floors were then adjusted iteratively so as to make χ2ν ≈ 1.0 for each coordinate. This procedure resulted in error floors of 0.08 and 0.16 mas for the eastward and northward position measurements, respectively. The magnitudes of these error floors are reasonably consistent with those obtained for other parallax targets observed at 8.4 GHz, e.g., Menten et al. (2007), and are probably dominated by unmodeled ionospheric delays. Any component of variability in the centroid of the core–jet position of Cygnus X-1 caused by changing jet opacity must be less than ≈0.1 mas."[10]

## Ions

"The very high electron column densities toward PSRs B0823+26 (0.055 cm−3), B0833−45 (0.270 cm−3), B0950+08 (0.023 cm−3) and J1024−0719 (ne > 0.029 cm−3) in quadrant 3 indicate the presence of dense ionized gas immediately beyond the LHB."[9]

"If the dense ionized material is outside the LHB, the relative deficiency of electrons along the LOS to PSR J0437−4715 may be accounted for if the gas is clumped or has a non-uniform z-distribution. There is independent evidence for the existence of ionized clouds in this region."[9]

"HII with ne = 0.07-0.14 cm−3 fills 40-90 pc of the β CMa LOS. [...] two clouds dominate this LOS."[9]

## Sun

"During the twentieth century, Eros data were used to determine various values for the ratio of the sun’s mass to that of the Earth-moon system. Observations of Eros were also well suited for determining the value of the astronomical unit using either the trigonometric parallax method or the dynamical method. The solar parallax is defined as the angle subtended by the Earth’s radius as seen from a distance of 1 AU (about 8.8 arc seconds). As an object like Eros closely approaches the Earth, its apparent position on the plane-of-sky is very sensitive to the observer’s location on the Earth’s surface so that position observations can be used to solve for the value of solar parallax; given the radius of the Earth (in km), the AU is then determined. The dynamical method of determining the AU depends upon using the astrometric observations of a close Earth approaching object to determine the system mass of the Earth and moon. The mass of the Earth-moon system is directly related to the solar parallax value through a combination of Kepler’s third law and the equation of the acceleration of gravity at a given geocentric distance. An interesting history of the various trigonometric and dynamical attempts to determine the solar parallax is given by Eugene Rabe [3]. However, with the use of radar to observe the planets, the value of the AU has been refined beyond the accuracy possible using observations of close Earth approaching asteroids."[29]

## Earth

"Diurnal parallax is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars."[30][31]

## Moon

Lunar parallax (often short for lunar horizontal parallax or lunar equatorial horizontal parallax), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.[32]

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:- one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram); and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth[33] -- equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its perturbed and approximately elliptical orbit around the Earth. The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.[32] The Astronomical Almanac and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of lunar theory.

Parallax can also be used to determine the distance to the Moon.

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000 km. This procedure was first used by Aristarchus of Samos[34] and Hipparchus, and later found its way into the work of Ptolemy. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:

${\displaystyle \mathrm {distance} _{\textrm {moon}}={\frac {\mathrm {distance} _{\mathrm {observerbase} }}{\tan(\mathrm {angle} )}}}$

Any distance to the Moon is often initially calculated as a multiple of the Earth radius ${\displaystyle R_{\oplus }}$ .

## Hipparcos

Hipparcos is the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial objects.

These measurements allow the accurate determination of proper motions and parallaxes of stars, their distance and tangential velocity.

## Hubble Space Telescope

"Using NASA's Hubble Space Telescope, astronomers now can precisely measure the distance of stars up to 10,000 light-years away -- 10 times farther than previously possible."[35]

"Astronomers have developed yet another novel way to use the 24-year-old space telescope by employing a technique called spatial scanning, which dramatically improves Hubble's accuracy for making angular measurements. The technique, when applied to the age-old method for gauging distances called astronomical parallax, extends Hubble's tape measure 10 times farther into space."[35]

"By applying a technique [illustrated in the image at the right] called spatial scanning to an age-old method for gauging distances called astronomical parallax, scientists now can use NASA’s Hubble Space Telescope to make precision distance measurements 10 times farther into our galaxy than previously possible."[35]

"This new capability is expected to yield new insight into the nature of dark energy, a mysterious component of space that is pushing the universe apart at an ever-faster rate."[36]

"Parallax, a trigonometric technique, is the most reliable method for making astronomical distance measurements, and a practice long employed by land surveyors here on Earth. The diameter of Earth's orbit is the base of a triangle and the star is the apex where the triangle's sides meet. The lengths of the sides are calculated by accurately measuring the three angles of the resulting triangle."[35]

"Astronomical parallax works reliably well for stars within a few hundred light-years of Earth. For example, measurements of the distance to Alpha Centauri, the star system closest to our sun, vary only by one arc second. This variance in distance is equal to the apparent width of a dime seen from two miles away."[35]

"Stars farther out have much smaller angles of apparent back-and-forth motion that are extremely difficult to measure. Astronomers have pushed to extend the parallax yardstick ever deeper into our galaxy by measuring smaller angles more accurately."[35]

"This new long-range precision was proven when scientists successfully used Hubble to measure the distance of a special class of bright stars called Cepheid variables, approximately 7,500 light-years away in the northern constellation Auriga. The technique worked so well, they are now using Hubble to measure the distances of other far-flung Cepheids."[35]

"Such measurements will be used to provide firmer footing for the so-called cosmic "distance ladder." This ladder's "bottom rung" is built on measurements to Cepheid variable stars that, because of their known brightness, have been used for more than a century to gauge the size of the observable universe. They are the first step in calibrating far more distant extra-galactic milepost markers such as Type Ia supernovae."[35]

"To make a distance measurement, two exposures of the target Cepheid star were taken six months apart, when Earth was on opposite sides of the sun. A very subtle shift in the star's position was measured to an accuracy of 1/1,000 the width of a single image pixel in Hubble's Wide Field Camera 3, which has 16.8 megapixels total. A third exposure was taken after another six months to allow for the team to subtract the effects of the subtle space motion of stars, with additional exposures used to remove other sources of error."[35]

## Nano-JASMINE

Nano-JASMINE is the Nano-Japan Astrometry Satellite Mission for INfrared Exploration.

"Nano-JASMINE is a 50cm class micro satellite that has space astrometry mission for the first time in Japan. Making a map of many stars, Nano-JASMINE will take us a knowledge of our Galaxy, and techniques of observation. Intelligent Space Systems Laboratory (the University of Tokyo) that took two CubeSats and one Micro-Satellite into orbit covers bus system, and National Astronomical Observatory of Japan (NAOJ) that plans more precise missions by larger satellites covers mission telescope."[37]

## Gaia spacecraft

"Gaia [NSSDC/COSPAR ID: 2013-074A] is a European Space Agency astronomy mission whose primary goals are to: (1) measure the positions and velocity of approximately one billion stars; (2) determine the brightness, temperature, composition, and motion through space of those stars; and, (3) create a three-dimensional map of the Milky Way galaxy."[38]

At left is an image of the Milky Way galaxy depicted onto it the various targets and experimental regions for the ESA Gaia spacecraft.

"Repeatedly scanning the sky, Gaia will observe each of the billion stars an average of 70 times each over the five years. It will measure the position and key physical properties of each star, including its brightness, temperature and chemical composition."[39]

At lower right is an image from an animation that has the Gaia spacecraft spinning slowly (four revolutions per day) to sweep its two telescopes across the entire celestial sphere.

Diagram of Gaia
Mirrors (M)
• Mirrors of telescope 1 (M1, M2 and M3)
• Mirrors of telescope 2 (M'1, M'2 and M'3)
• mirrors M4, M'4, M5, M6 are not shown
Other components (1–9)
1. Optical bench (silicon carbide torus)
3. Focal plane electronics[40]
4. Nitrogen tanks
5. Diffraction grating spectroscope
6. Liquid propellant tanks
7. Star trackers
8. Telecommunication panel and batteries
9. Main propulsion subsystem
(A) Light path of telescope 1
Design of the focal plane and instruments

The design of the Gaia focal plane and instruments. Due to the spacecraft's rotation, images cross the focal place array right-to-left at 60 arc seconds per second.[40]

1. Incoming light from mirror M3
2. Incoming light from mirror M'3
3. Focal plane, containing the detector for the Astrometric instrument in light blue, Blue Photometer in dark blue, Red Photometer in red, and Radial Velocity Spectrometer in pink.
4. Mirrors M4 and M'4, which combine the two incoming beams of light
5. Mirror M5
6. Mirror M6, which illuminates the focal plane
7. Optics and diffraction grating for the Radial Velocity Spectrometer (RVS)
8. Prisms for the Blue Photometer and Red Photometer (BP and RP)

## Hypotheses

1. In order for X-ray trigonometric parallax measurements to succeed we either need a much larger spatial width for observing X-ray source apparent movement or try to incorporate

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