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The model for the declination is shown 1 Fig.
The model for the declination is shown in Fig.
17 (Right).
\ref{0605_precession_model} (Right).
We found another set of parameters Q=92.27, dp=69.87. Yo=12.9" as one of the possible solutions.
We found another set of parameters $\Omega = 92.2^\circ$, $\phi_{0} = 69.8^\circ$, $\eta_{0} = 12.9^\circ$ as one of the possible solutions.
However. the viewing angle is very large.
However, the viewing angle is very large.
Using the equation and taking average value of apparent speeds Bapp.arer16.5c. we can calculate the speed in the source frame 2.49c. which is higher then the speed of light.
Using the equation and taking average value of apparent speeds $\beta_{app,aver} = 16.5c$ , we can calculate the speed in the source frame $\beta = 2.49c$ , which is higher then the speed of light.
Precession cone aperture angle Q=92.2° is also unlikely.
Precession cone aperture angle $\Omega = 92.2^\circ$ is also unlikely.
Therefore. this set of parameters is not plausible.
Therefore, this set of parameters is not plausible.
The solution in Table 6 is the only physically feasible. since all the parameter space was covered during the fitting procedure.
The solution in Table \ref{0605_precession} is the only physically feasible, since all the parameter space was covered during the fitting procedure.
Assuming that the helical motion of the quasi-stationary jet component Cl is caused solely by jet precession (with a period of 8 years). we were able to determine the geometrical parameters of the jet precession. such as the aperture angle of à precession cone. the viewing angle. and the projection angle.
Assuming that the helical motion of the quasi-stationary jet component $C1$ is caused solely by jet precession (with a period of 8 years), we were able to determine the geometrical parameters of the jet precession, such as the aperture angle of a precession cone, the viewing angle, and the projection angle.
The fitted viewing angle όν=2.6°+2.2° agrees well with the upper limit for the viewing angle deduced from the apparent speeds of the jet components oj,=3.7°.
The fitted viewing angle $\phi_{0} = 2.6^\circ \pm 2.2^\circ$ agrees well with the upper limit for the viewing angle deduced from the apparent speeds of the jet components $\phi_\mathrm{max} = 3.7^\circ$.
It is also similar to the viewing angle (4,=5° derived from the radio total flux-density variability by Hovatta et al. (
It is also similar to the viewing angle $\phi_\mathrm{var} = 5^\circ$ derived from the radio total flux-density variability by Hovatta et al. (
2008).
2008).
The Jet precession will also cause the periodical variability of the radio total flux-density light curves with the same period of ὃ years. due to periodical changes of the Doppler factor.
The jet precession will also cause the periodical variability of the radio total flux-density light curves with the same period of 8 years, due to periodical changes of the Doppler factor.
The flux density 5; is changing with the Doppler factor due to the jet precession ο=500.y). where « is the spectral index (Lind Blandford 1985).
The flux density $S_{j}$ is changing with the Doppler factor due to the jet precession $S_{j} = S_{j}' \delta (\phi, \gamma)^{p+\alpha}$, where $\alpha$ is the spectral index (Lind Blandford 1985).
From this equation we can expect that the flares due to the jet precession will appear simultaneously at different frequencies. which is actually observed for the major outbursts in. 1973. 1988. and 1995-1996.
From this equation we can expect that the flares due to the jet precession will appear simultaneously at different frequencies, which is actually observed for the major outbursts in 1973, 1988, and 1995-1996.
Therefore. the jet precession 1s probably responsible for the total flux-density variability. periodicity in the outbursts. and the helical motion of the quasi-stationary Jet component C1.Some questions remam unclear.
Therefore, the jet precession is probably responsible for the total flux-density variability, periodicity in the outbursts, and the helical motion of the quasi-stationary jet component $C1$ .Some questions remain unclear.
Itis still not known why the four major outbursts in the total flux-density light curve which repeat periodically with an 8-year period have different spectral properties.
Itis still not known why the four major outbursts in the total flux-density light curve which repeat periodically with an 8-year period have different spectral properties.
Longer monitoring of this quasar at several frequencies is necessary in order to understand spectral properties of the fares.
Longer monitoring of this quasar at several frequencies is necessary in order to understand spectral properties of the flares.
The precession of BOGOS—O8S5 jet can be caused by several physical mechanisms.
The precession of $-$ 085 jet can be caused by several physical mechanisms.
Jet instabilities. a secondary black holes
Jet instabilities, a secondary black holes
is limited by £.
is limited by $\xi$.
LU the inclination of a scan circle with respect o the ecliptic plane is given by y (with »=2/2 lor a scan circle crossing the ecliptic plane at right angles) then where à is the opening-anele of the detector as defined above in Section 2.2..
If the inclination of a scan circle with respect to the ecliptic plane is given by $\varphi$ (with $\varphi=\pm\pi/2$ for a scan circle crossing the ecliptic plane at right angles) then where $\alpha$ is the opening-angle of the detector as defined above in Section \ref{sec:projeffects}.
Thus. for a small solar aspect angle. he inclinations of the scan circles remain all close to »erpendieular to the ecliptic plane.
Thus, for a small solar aspect angle, the inclinations of the scan circles remain all close to perpendicular to the ecliptic plane.
This is a relatively unfavourable situation for point- ancl compact-source analysis as the easier to extract positional information is in the along-scan direction.
This is a relatively unfavourable situation for point- and compact-source analysis as the easier to extract positional information is in the along-scan direction.
“Phis is therefore a good configuration to see how the current analysis will perform under dillieult conditions.
This is therefore a good configuration to see how the current analysis will perform under difficult conditions.
For missions such as Lipparcos and Gaia. which aimi specifically at the astrometry. of point sources. the Solar-aspect angle. has been maximized.. at values of 43 and 45 degrees respectively (77)... while for Planck it has been chosen at a relatively small value of 7.5 degrees (2)..
For missions such as Hipparcos and Gaia, which aim specifically at the astrometry of point sources, the Solar-aspect angle has been maximized, at values of 43 and 45 degrees respectively \citep{fvl97,lindeg08}, while for Planck it has been chosen at a relatively small value of 7.5 degrees \citep{dupac05}.
Phe Planck mission has therefore been chosen as an example case for our experiments. as it provides a wide range of conditions depending on the ecliptic Latitude of the source.
The Planck mission has therefore been chosen as an example case for our experiments, as it provides a wide range of conditions depending on the ecliptic latitude of the source.
The precession period for the spin axis of the Planck satellite is scheduled to be 6 months.
The precession period for the spin axis of the Planck satellite is scheduled to be 6 months.
The Planck scanning (7) is described. not as a continuous function. as for Hipparcos ancl Gaia. but. step-wise.
The Planck scanning \citep{dupac05} is described not as a continuous function, as for Hipparcos and Gaia, but step-wise.
This means that the spin axis remains at a κος position over a period of. for example. 1 hour. during which the satellite describes 60. full circles.
This means that the spin axis remains at a “fixed" position over a period of, for example, 1 hour, during which the satellite describes 60 full circles.
The. nominal. scan velocity is 6 degrees per second. and every. part of the scan is examined by GO successive scans.
The nominal scan velocity is 6 degrees per second, and every part of the scan is examined by 60 successive scans.
The accumulated. data from the scan circles are referred to as a “ring”.
The accumulated data from the scan circles are referred to as a “ring".
Due to the time required for repositioning the spin axis. there are ellectively only about 55 uscable scans per pointing.
Due to the time required for repositioning the spin axis, there are effectively only about 55 useable scans per pointing.
Scans in successive pointings will have a maximum separation of about 2.5£0.65 arcmin.
Scans in successive pointings will have a maximum separation of about $2.5\pm0.65$ arcmin.
The mean value is the mean motion of the Sun on the ecliptic. the range is the result of precession of the spin axis around the Sun. as a result of which the scan density will be higher when the spin axis moves in opposite direction of the Sun and vice versa.
The mean value is the mean motion of the Sun on the ecliptic, the range is the result of precession of the spin axis around the Sun, as a result of which the scan density will be higher when the spin axis moves in opposite direction of the Sun and vice versa.
For our experiments we have used one vear of mission data. providing a generally homogeneous scan coverage of the sky.
For our experiments we have used one year of “mission data”, providing a generally homogeneous scan coverage of the sky.
For testing the methods described in Section 2. we used a “worst-case” scenario bv selecting the 30 Gllz detectors. which have the largest beam size (EWIIM-33 arcmin) and the largest. predicted beam ellipticity. (1.43).
For testing the methods described in Section \ref{sec:technique} we used a “worst-case" scenario by selecting the 30 GHz detectors, which have the largest beam size $=33$ arcmin) and the largest predicted beam ellipticity (1.4).
For this detector we adapted an opening-angle of 85 degrees. (this is for testing purposes only ancl doesn't represent the actual detector position in the Planck focal plane).
For this detector we adapted an opening-angle of 85 degrees (this is for testing purposes only and doesn't represent the actual detector position in the Planck focal plane).
At the sampling rate of 32.5 Hz. the samples are spaced 5.4 arcmin apart. and a point-source image is effectively covered by 6 to S samples.
At the sampling rate of 32.5 Hz, the samples are spaced 5.4 arcmin apart, and a point-source image is effectively covered by 6 to 8 samples.
The HAVE. (see section 2)) has been represented by a 2-dimensional Ciaussian distribution: where a=(0.5/V21n20.425 FWIIM.
The IRF (see section \ref{sec:technique}) ) has been represented by a 2-dimensional Gaussian distribution: where $\sigma=0.5/\sqrt{2\ln 2}\approx 0.425$ FWHM.
The shape of the IRE has only a very minor ellect on the results presented in this study.
The shape of the IRF has only a very minor effect on the results presented in this study.
The main effect on the positional accuracies comes from the image "width" and the steepness of its slopes. and a CGaussian. profile forms a. good. first approximation for a tvpical beam profile.
The main effect on the positional accuracies comes from the image “width" and the steepness of its slopes, and a Gaussian profile forms a good first approximation for a typical beam profile.
Two distributions of samples have been considered. one following a realistic scan-velocity. which will generally create an uneven distribution of samples over an image. and a "smooth" sampling. with all samples spread: evenly over the image.
Two distributions of samples have been considered, one following a realistic scan-velocity, which will generally create an uneven distribution of samples over an image, and a “smooth" sampling, with all samples spread evenly over the image.
The latter was finally preferred. for. testing. as it should. provide an simple-to-interpret set of. results. not unnecessarily complicating the possible detection. of systematic errors that may possibly be inherent to. the methods tested.
The latter was finally preferred for testing, as it should provide an simple-to-interpret set of results, not unnecessarily complicating the possible detection of systematic errors that may possibly be inherent to the methods tested.
Thus. at cach pointing direction a set of evenly spaced samples was created that represented the 60 scans mace at that pointing.
Thus, at each pointing direction a set of evenly spaced samples was created that represented the 60 scans made at that pointing.
Gaussian noise was added to these samples at a level according to a specified. signal to noise ratio. which applied to the peak of the image.
Gaussian noise was added to these samples at a level according to a specified signal to noise ratio, which applied to the peak of the image.
The contributing scans for each source were determined by the adopted scanning law. which defines the orientation anc position of cach scan relative to that of the source.
The contributing scans for each source were determined by the adopted scanning law, which defines the orientation and position of each scan relative to that of the source.
For each set of contributing transits of a given source. LOO dillerent noise realizations were mace.
For each set of contributing transits of a given source, 100 different noise realizations were made.
The available simulated data for cach source were investigated following the methods clescribecl in Section 2.4 through weighted least-squares solutions. providing estimated values. for the position ancl Εαν (the— image parameters). and the standard. deviations on those values.
The available simulated data for each source were investigated following the methods described in Section \ref{sec:correst} through weighted least-squares solutions, providing estimated values for the position and flux (the image parameters), and the standard deviations on those values.
‘These parameter values were compared with the input data. and the dilferences compared. with the estimated. standard errors.
These parameter values were compared with the input data, and the differences compared with the estimated standard errors.
The weights used in the solutions were based on the standard. deviation of the Gaussian noise used in the data simulations.
The weights used in the solutions were based on the standard deviation of the Gaussian noise used in the data simulations.
Phe mean values of the image parameters ancl their standard. errors were determined as based on the 100 independent. noise realizations.
The mean values of the image parameters and their standard errors were determined as based on the 100 independent noise realizations.
This allows in. principle for bias detection at LO per cent of the standard deviation.
This allows in principle for bias detection at 10 per cent of the standard deviation.
For a S/N=10. typical values of just under 2 aresee were founcl for the positional standard. error. with a sigma for the beam of S41 arcsec.
For a $S/N=10$, typical values of just under 2 arcsec were found for the positional standard error, with a sigma for the beam of 841 arcsec.
For the Dux. the standard error was approximately 0.15 per cent.
For the flux, the standard error was approximately 0.15 per cent.
The observed clispersion in the nmieasured values was in good agreement with the derived standard errors for the solutions.
The observed dispersion in the measured values was in good agreement with the derived standard errors for the solutions.
More details on results are provided in Section 6..
More details on results are provided in Section \ref{sec:comparisons}.
As part of the experiment. the samples as obtained in the scans were projected on a map of square pixels.
As part of the experiment, the samples as obtained in the scans were projected on a map of square pixels.
To avoid complications caused by. distorted: pixels. cach source was projected. as if it was situated at the ecliptic plane. where the map approaches a Dat patch.
To avoid complications caused by distorted pixels, each source was projected as if it was situated at the ecliptic plane, where the map approaches a flat patch.
The pixel size was chosen ab 6 arcmin. which is compatible with the sample size on the scans (5.4 arcmin) and the maximum distance between successive scans (3.15 arcmin). and guarantees that at least two samples are contributing to cach pixel.
The pixel size was chosen at 6 arcmin, which is compatible with the sample size on the scans (5.4 arcmin) and the maximum distance between successive scans (3.15 arcmin), and guarantees that at least two samples are contributing to each pixel.
“Phe 6 arcmin
The 6 arcmin
Currently more than 460 have been detected through various methods.
Currently more than 460 have been detected through various methods.
With improving instrument precision smaller and less massive objects are or will be soon accessible observationally.
With improving instrument precision smaller and less massive objects are or will be soon accessible observationally.
With current instruments like Kepler it should even be possible to detect moons of exoplanets (?????)..
With current instruments like Kepler it should even be possible to detect moons of exoplanets \citep{SS99,SSS07,SSS09,K09,KFC09}.
But as most planets are found by methods most sensitive to massive planets with a small semi-major axis (7Hot-Jupiters") the question arises not only if it is possible to detect exomoons but also how likely it is for them to form and survive in the first place.
But as most planets are found by methods most sensitive to massive planets with a small semi-major axis (“Hot-Jupiters”) the question arises not only if it is possible to detect exomoons but also how likely it is for them to form and survive in the first place.
Several studies (??) explore the stability of orbits around gas giants.
Several studies \citep{BO02,DWY06} explore the stability of orbits around gas giants.
In this contribution we apply to a sample of observed exoplanets the results of ? on the stability of moons around gas giants.
In this contribution we apply to a sample of observed exoplanets the results of \citet{DWY06} on the stability of moons around gas giants.
Our sample (Tab. 1))
Our sample (Tab. \ref{tab:exoplanets}) )
includes all published transiting exoplanets for which the mass and radius of the planet and the host star. and the orbital parameters are all reasonably well known.
includes all published transiting exoplanets for which the mass and radius of the planet and the host star, and the orbital parameters are all reasonably well known.
The region of orbital stability around a close-in gas-giant planet is set by two radii.
The region of orbital stability around a close-in gas-giant planet is set by two radii.
We assume that the smallest orbit is set by the Roche-radius.
We assume that the smallest orbit is set by the Roche-radius.
Any moon larger than a few km within the Roche-limit of its planet would be torn apart by the tidal forces between the planet and the moon.
Any moon larger than a few km within the Roche-limit of its planet would be torn apart by the tidal forces between the planet and the moon.
The Roche-radius. Aj. depends mainly on the density of the two interacting objects and can be written forfluid-like objects as (2):: where rp) is the radius of the planet. p, and p, are the mean densities of the planet and the moon. and z5 the mass of the planet.
The Roche-radius, $R_\mathrm{roche}$, depends mainly on the density of the two interacting objects and can be written forfluid-like objects as \citep{BT87}: where $r_\mathrm{p}$ is the radius of the planet, $\rho_\mathrm{p}$ and $\rho_\mathrm{m}$ are the mean densities of the planet and the moon, and $m_\mathrm{p}$ the mass of the planet.
Of course the Roche criterion also limits the minimal semi-major axis. a. of the planet's orbit around its star.
Of course the Roche criterion also limits the minimal semi-major axis, $a$, of the planet's orbit around its star.
The second part of eq.
The second part of eq.
| shows that Ryoche is independent of rp.
\ref{eq:roche} shows that $R_\mathrm{roche}$ is independent of $r_\mathrm{p}$.
The outer limit for stable orbits of à moon around an exoplanet is the so-called Hill-radius. which defines the sphere in which the gravitational pull of the planet on the moon is larger than that of the star.
The outer limit for stable orbits of a moon around an exoplanet is the so-called Hill-radius, which defines the sphere in which the gravitational pull of the planet on the moon is larger than that of the star.
The Hill-radius ts given as (?):: where M. is the mass of the star.
The Hill-radius is given as \citep{B86}: where $M_\mathrm{\ast}$ is the mass of the star.
By using numerical integrations of the equations of motion. recent studies (e.g.?) found that the Hill-radius over estimates the maximum stable orbital radius by a factor f.
By using numerical integrations of the equations of motion, recent studies \citep[e.g.][]{BO02} found that the Hill-radius over estimates the maximum stable orbital radius by a factor $f$.
? studied this question in detail and derived two equations for the maximal stable orbital radi one for prograde motion of the moon and the other one for retrograde motion.
\citet{DWY06} studied this question in detail and derived two equations for the maximal stable orbital radii, one for prograde motion of the moon and the other one for retrograde motion.
Both depend on the eccentricities. ey. for the planet's orbit and e, for the moon’s.
Both depend on the eccentricities, $e_\mathrm{p}$, for the planet's orbit and $e_\mathrm{m}$ for the moon's.
For a prograde satellite ? give: and for retrograde ones: ? also studied the possible lifetime of a moon due to orbital decay as a result of tidal dissipation of angular momentum.
For a prograde satellite \citet{DWY06} give: and for retrograde ones: \citet{BO02} also studied the possible lifetime of a moon due to orbital decay as a result of tidal dissipation of angular momentum.
Based on this result ? also derived an equation. for the maximum mass à moon can have for a given distance to the planet: Q, is the dimensionless tidal dissipation factor. kop the tidal Love number. 7 the moon's lifetime and fRy either the pro- or the retrograde Hill radius.
Based on this result \citet{DWY06} also derived an equation for the maximum mass a moon can have for a given distance to the planet: $Q_\mathrm{p}$ is the dimensionless tidal dissipation factor, $k_\mathrm{2P}$ the tidal Love number, $T$ the moon's lifetime and $f\,R_\mathrm{Hill}$ either the pro- or the retrograde Hill radius.
Q) is very poorly constrained even for the planets of the solar system and even more uncertain for exoplanets.
$Q_\mathrm{p}$ is very poorly constrained even for the planets of the solar system and even more uncertain for exoplanets.
Following ? we chose Q, = 10° and ορ = 0.51.
Following \citet{BO02} we chose $Q_\mathrm{p}$ = $^5$ and $k_\mathrm{2P}$ = 0.51.
Por the satellite lifetime. 7. we adopt the minimum age of the parental star if given.
For the satellite lifetime, $T$, we adopt the minimum age of the parental star if given.
For stars lacking age determinations. a minimum age of | Gyr is used.
For stars lacking age determinations, a minimum age of 1 Gyr is used.
We consider moon/planet mass ratios q< 0.1. though. the tidal effects on the planetary rotation by moons with g> 0.01 might already modify the result (?)..
We consider moon/planet mass ratios $q \le$ 0.1, though, the tidal effects on the planetary rotation by moons with $q >$ 0.01 might already modify the result \citep{BO02}. .
Such cases are marked with a « in Table I..
Such cases are marked with a $d$ ' in Table \ref{tab:exoplanets}.
The largest uncertainty in eq.
The largest uncertainty in eq.
5. lies in Qu.
\ref{eq:moonmax} lies in $Q_\mathrm{p}$ .
While a Q,- 10° is commonly used. ? suggestec values as high as 10" for exoplanets.
While a $Q_\mathrm{p} \sim$ $^5$ is commonly used, \citet{CMA09} suggested values as high as $^{13}$ for exoplanets.
A recent study (?) derived ο = 3.6 x10* for Jupiter throughastrometric observations of the planet and its moon Io.
A recent study \citep{LAK09} derived $Q_\mathrm{p}$ = 3.6 $\times10^4$ for Jupiter throughastrometric observations of the planet and its moon Io.
In this context it is interesting to note (2) that
In this context it is interesting to note \citep{BO02} that
profile is nearly time-independent.
profile is nearly time-independent.
It is clear from Figure 2. that the cluster can be divided into three regimes in radius.
It is clear from Figure \ref{fig_rho} that the cluster can be divided into three regimes in radius.
In the outer region (r>20 kpc), where the NFW dark matter dominates the gravitational potential, the gas properties stay relatively constant during the simulation.
In the outer region $r>20$ kpc), where the NFW dark matter dominates the gravitational potential, the gas properties stay relatively constant during the simulation.
Inside the core, but outside the transition radius (0.1 kpc Sr20 kpc), where the gravitational potential is dominated by the BCG, the density and pressure increase slowly while the temperature profile is nearly flat.
Inside the core, but outside the transition radius $0.1$ kpc $\lesssim r\lesssim 20$ kpc), where the gravitational potential is dominated by the BCG, the density and pressure increase slowly while the temperature profile is nearly flat.
Finally, in the very center of the cluster (r<0.1 within the radius of influence of the SMBH where kpc),the gravity is dominated by the SMBH (50 pc, see Figure 1)), the cooling catastrophe first occurs at small radius.
Finally, in the very center of the cluster $r \lesssim 0.1$ kpc), within the radius of influence of the SMBH where the gravity is dominated by the SMBH $\sim 50$ pc, see Figure \ref{fig_g}) ), the cooling catastrophe first occurs at small radius.
Then the transition radius moves slowly outwards with time, from ~10 pc when the collapse first happens to about ~100 pc, about 10 Myr later.
Then the transition radius moves slowly outwards with time, from $\sim 10$ pc when the collapse first happens to about $\sim 100$ pc, about $10$ Myr later.