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the contour plots presented in Figs.
the contour plots presented in Figs.
3 - 5 have been calculated first without the imposition of the physical constraints discussed in Section 2.
\ref{fig3} - \ref{fig5} have been calculated first without the imposition of the physical constraints discussed in Section 2.
A typical example of the application of these conditions, and their impact on the credible regions, is shown in Fig.
A typical example of the application of these conditions, and their impact on the credible regions, is shown in Fig.
7 for the acoustic scale, la.
\ref{fig7} for the acoustic scale, $l_a$.
One can see in Fig. 7((
One can see in Fig. \ref{fig7}( (
b) how the contours are abruptly cut off, in this case by the inclusion of the condition that the universe currently be accelerating.
b) how the contours are abruptly cut off, in this case by the inclusion of the condition that the universe currently be accelerating.
In other words, in the region enclosed by the contours in the right hand part of Fig. 7((
In other words, in the region enclosed by the contours in the right hand part of Fig. \ref{fig7}( (
a), e.g. for yo«0.6 and n» 1.5, our SFS model predicts a value of J, which is in satisfactory agreement with observations, but for a universe which is not currently accelerating.
a), e.g. for $y_0 < 0.6$ and $n > 1.5$ , our SFS model predicts a value of $l_a$ which is in satisfactory agreement with observations, but for a universe which is not currently accelerating.
The other cosmological probes such as the age of the universe and the BAO distance parameter are similarly affected, although much less severely, for certain values of 6.
The other cosmological probes such as the age of the universe and the BAO distance parameter are similarly affected, although much less severely, for certain values of $\delta$.
However, since it is already the case that we find no significant overlap between the credible regions over the entire parameter space applying these additional physical constraints, we will not present any further plots that do include them.
However, since it is already the case that we find no significant overlap between the credible regions over the entire parameter space applying these additional physical constraints, we will not present any further plots that do include them.
In this paper we have investigated one class of Sudden Future Singularity models proposed by Barrow, by confronting it with the currently available observational data.
In this paper we have investigated one class of Sudden Future Singularity models proposed by Barrow, by confronting it with the currently available observational data.
After introducingthe theory behind the model and
After introducingthe theory behind the model and
2010).
.
The radial Coriolis force may drive the meridional circulation in low and perhaps midlatitudes, but it can not do so in high latitudes, because, since there the rotation axis and the local vertical are nearly parallel, the radial Coriolis force is very weak (see Figure 2).
The radial Coriolis force may drive the meridional circulation in low and perhaps midlatitudes, but it can not do so in high latitudes, because, since there the rotation axis and the local vertical are nearly parallel, the radial Coriolis force is very weak (see Figure 2).
Moreover, retaining it in the equations of motion complicates the problem mathematically.
Moreover, retaining it in the equations of motion complicates the problem mathematically.
In particular, retaining the radial component of Coriolis force precludes using separation of variables to solve even the axisymmetric problem for a spherical polar cap.
In particular, retaining the radial component of Coriolis force precludes using separation of variables to solve even the axisymmetric problem for a spherical polar cap.
Therefore we will begin from the spherical shell equations but then approximate the spherical polar cap (see Figure 3) by a cylinder with the same polar axis; this eliminates the relatively mild
Therefore we will begin from the spherical shell equations but then approximate the spherical polar cap (see Figure 3) by a cylinder with the same polar axis; this eliminates the relatively mild
The transit probability will be as per Equation (11)).
The transit probability will be as per Equation \ref{eq:630}) ).
The angle @ is determined by the orbital separation of planet e and the host star at the time of their conjunction. (.2aresin(R./r,).
The angle $\theta_\mathrm{c}$ is determined by the orbital separation of planet `c' and the host star at the time of their conjunction, $\theta_\mathrm{c} = \arcsin(R_*/r_\mathrm{c})$.
As an example. consider the HAT-P-13 system (Bakoset 2009).
As an example, consider the HAT-P-13 system \citep{bakos2009}.
As of this only the inner planet. HAT- has been observed to transit.
As of this only the inner planet, HAT-P-13b, has been observed to transit.
What is the probability that the outer planet.
What is the probability that the outer planet.
HAT-P-13c. also transits?
HAT-P-13c, also transits?
Bakosetal.(2009) measure the orbital inclination of the inner planet to be iy283470.67.
\cite{bakos2009} measure the orbital inclination of the inner planet to be $i_{\mathrm{b,m}} = 83.4^\circ \pm 0.6^\circ$.
At the time of its conjunction with HAT-P-13. the outer planet is at a distance of r/R.=82.146.1 stellar radii from the star.
At the time of its conjunction with HAT-P-13, the outer planet is at a distance of $r_\mathrm{c}/R_* = 82.1 \pm 6.1$ stellar radii from the star.
This allows us to determine fjpA) and 0 for the HAT-P-13 system.
This allows us to determine $f_{I_\mathrm{b}}(i_\mathrm{b}|i_{\mathrm{b,m}})$ and $\theta_\mathrm{c}$ for the HAT-P-13 system.
For the mutual inclination of the two planets. we assumed that the orbital inclination of planet "ο was evenly distributed within Apemay degrees of the orbital inclination 4, of planet "b.
For the mutual inclination of the two planets, we assumed that the orbital inclination of planet `c' was evenly distributed within $\pm \lambda_{\mathrm{bc,max}}$ degrees of the orbital inclination $i_\mathrm{b}$ of planet `b'.
Figure 4 shows the transit probability of HAT-P-13e às a function of Ap,as.
Figure 4 shows the transit probability of HAT-P-13c as a function of $\lambda_{\mathrm{bc,max}}$.
The probability that planet c? transits is very dependent upon what we assume is a reasonable range of mutual inclination in the HAT-P-13 system.
The probability that planet `c' transits is very dependent upon what we assume is a reasonable range of mutual inclination in the HAT-P-13 system.
For reference. all of the Solar System planets are within 3.4° of the Earth’s orbit except for Mercury (at 7).
For reference, all of the Solar System planets are within $3.4^\circ$ of the Earth's orbit --- except for Mercury (at $7^\circ$ ).
If the mutual inclination of the two planets orbiting HAT-P-13 is within 3.47. then the outer planet will not transit.
If the mutual inclination of the two planets orbiting HAT-P-13 is within $3.4^\circ$, then the outer planet will not transit.
If the two planets are misaligned by up to 7. then the transit probability for planet c is7%.
If the two planets are misaligned by up to $7^\circ$, then the transit probability for planet `c' is.
. The maximum transit probability of occurs if we assume that the two planets may be inclined within 87 of each other.
The maximum transit probability of occurs if we assume that the two planets may be inclined within $8^\circ$ of each other.
As the assumed spread in mutual inclination increases. the transit probability will fall back to the a priori value of1.
As the assumed spread in mutual inclination increases, the transit probability will fall back to the a priori value of.
2%.. The transit probability for HAT-P-13e is therefore at most8.
The transit probability for HAT-P-13c is therefore at most.
5%. We now demonstrate how using stellar. inclination measurements and the enhanced transit probabilities (Section 2) can aid in the target selection of transit surveys.
We now demonstrate how using stellar inclination measurements and the enhanced transit probabilities (Section 2) can aid in the target selection of transit surveys.
As illustrative cases. we will calculate how many stars need to be observed in a survey looking for hot Jupiters. and for a separate survey searching for planets within the habitable-zone.
As illustrative cases, we will calculate how many stars need to be observed in a survey looking for hot Jupiters, and for a separate survey searching for planets within the habitable-zone.
In these examples. we will make the simplifying assumption that every star has either a hot Jupiter or habitable-zone planet in orbit. at distances of R..fa=1/10 or 1/215. respectively.
In these examples, we will make the simplifying assumption that every star has either a hot Jupiter or habitable-zone planet in orbit, at distances of $R_*/a=1/10$ or $R_*/a=1/215$ , respectively.
We assume that the inclinations of the planetary systems are distributed in two ways.
We assume that the inclinations of the planetary systems are distributed in two ways.
For the hot Jupiters. we use the planetary inclination distribution determined by Fabrycky&Winn(2009) from an ensemble of 11 Rossiter—MecLaughlin measurements of spin—orbit alignment.
For the hot Jupiters, we use the planetary inclination distribution determined by \cite{fabrycky2009} from an ensemble of 11 Rossiter—McLaughlin measurements of spin—orbit alignment.
The authors found that aside from the XO-3 the hot Jupiters they considered had planetary inclinations distributed according to a Rayleigh distribution with a width parameter of 6.67.
The authors found that aside from the XO-3 the hot Jupiters they considered had planetary inclinations distributed according to a Rayleigh distribution with a width parameter of $6.6^\circ$.
Exoplanets within the habitable-zone may not follow this same planetary inclination distribution.
Exoplanets within the habitable-zone may not follow this same planetary inclination distribution.
We use a uniform distribution of planetary inclination within 7.5° of the stellar equator, Earth has an planetary inclination of 7.155" to the Sun's equator,
We use a uniform distribution of planetary inclination within $7.5^\circ$ of the stellar equator; Earth has an planetary inclination of $7.155^\circ$ to the Sun's equator.
Including either spread of planetary inclinations into the calculations of transit probabilities acts to spread out the probability of transit. and make stars with stellar inclinations far from 90° more likely to show transits.
Including either spread of planetary inclinations into the calculations of transit probabilities acts to spread out the probability of transit, and make stars with stellar inclinations far from $90^\circ$ more likely to show transits.
At the same time. the spread of planetary inclinations makes stars with measured stellar inclinations near 907 less likely to show transits.
At the same time, the spread of planetary inclinations makes stars with measured stellar inclinations near $90^\circ$ less likely to show transits.
Assuming a measurement of (+,=907+5°. and that planetary inclinations are uniformly. distributed within 7.5" of the stellar equator. then the transit probability for a habitable-zone planet at a distance of R../a=1/215 star drops from to as compared to assuming the orbit is coplanar with the stellar equator.
Assuming a measurement of $\psi_m=90^\circ \pm 5^\circ$, and that planetary inclinations are uniformly distributed within $7.5^\circ$ of the stellar equator, then the transit probability for a habitable-zone planet at a distance of $R_*/a = 1/215$ star drops from to as compared to assuming the orbit is coplanar with the stellar equator.
Conversely. a star with measured stellar inclination of (0,=807+5” has its transit probability increased from to As our first example. take a survey for hot Jupiters around solar-type stars.
Conversely, a star with measured stellar inclination of $\psi_m=80^\circ \pm 5^\circ$ has its transit probability increased from to As our first example, take a survey for hot Jupiters around solar-type stars.
We will require a probability that the survey detects at least a single transiting planet.
We will require a probability that the survey detects at least a single transiting planet.
From Equation (17)) this means that P44=0.95.
From Equation \ref{eq:2540}) ) this means that $P_\mathrm{det}=0.95$.
Although we may arbitrarily set Pop, and 7. as shown in the appendix the time required to complete a hot Jupiter survey is minimized if we set Pa,=0.9814.
Although we may arbitrarily set $P_\mathrm{obs}$ and $n_\mathrm{i}$, as shown in the appendix the time required to complete a hot Jupiter survey is minimized if we set $P_\mathrm{obs}=0.9814$.
To find the number of stars needed in the initial sample we must then solve We must therefore have jj=29 stars in our initial sample.
To find the number of stars needed in the initial sample we must then solve We must therefore have $n_\mathrm{i}=29$ stars in our initial sample.
We next want to know how many stars out of these 29 will actually have to be observed photometrically.
We next want to know how many stars out of these 29 will actually have to be observed photometrically.
That is. how many of the mitial targets with measured inclinations near 90° will we need to look at for transits?
That is, how many of the initial targets with measured inclinations near $90^\circ$ will we need to look at for transits?
We will assume that all of the stellar inclination measurementshave Gaussian uncertainties of 5°.
We will assume that all of the stellar inclination measurementshave Gaussian uncertainties of $5^\circ$.
The transit probability for a hot Jupiter can be calculated for various orientation measurements of the form c,d:5" by using Equation (10)) to account for the inclination of the planetary system: hot Jupiters at a distance ofR.fa=1/10 will show transits up to à maximum angle of 025.737.
The transit probability for a hot Jupiter can be calculated for various orientation measurements of the form $\psi_m \pm 5^\circ$ by using Equation \ref{eq:620}) ) to account for the inclination of the planetary system: hot Jupiters at a distance of $R_*/a = 1/10$ will show transits up to a maximum angle of $\theta = 5.73^\circ$.
The angle o that defines our observed subsample solves and is o=24.017 To detect at least one hot Jupiter. we must therefore photometrically observe stars that will have measured stellar inclinations within 24.01" of 907.
The angle $\phi$ that defines our observed subsample solves and is $\phi=24.01^\circ$ To detect at least one hot Jupiter, we must therefore photometrically observe stars that will have measured stellar inclinations within $24.01^\circ$ of $90^\circ$.
This will give us a probability of Py=95% of detecting at least one hot Jupiter.
This will give us a probability of $P_\mathrm{det}=95$ of detecting at least one hot Jupiter.
The top panel of Figure 5 shows how the number of stars that need to be observed varies as a function of the stellar inclination measurement precision for various confidence levels.
The top panel of Figure 5 shows how the number of stars that need to be observed varies as a function of the stellar inclination measurement precision for various confidence levels.
The top panel also shows how the fraction of the mitial target list that will need to be observed varies with measurement precision.
The top panel also shows how the fraction of the initial target list that will need to be observed varies with measurement precision.
The lower limits in both cases set by the spread in the distribution of f(A) as determined by Fabrycky&Winn (2009).
The lower limits in both cases set by the spread in the distribution of $f_\Lambda(\lambda)$ as determined by \cite{fabrycky2009}. .
In a notional survey for habitable-zone planets. we will also require a chance of detecting at least. one transit.
In a notional survey for habitable-zone planets, we will also require a chance of detecting at least one transit.
The calculations are similar to those for the survey
The calculations are similar to those for the survey
the ACT results from and SPT results from (2011).
the ACT results from and SPT results from .
ACT find a
ACT find a
have a radial distribution proportional to radius. aud should be found in Che vicinity of ry. nol concentrated within 0.27, as is predicted under the black hole conjecture.
have a radial distribution proportional to radius, and should be found in the vicinity of $r_h$, not concentrated within $0.2r_h$ as is predicted under the black hole conjecture.
Fast-movine stars can have origins other than a black hole. of course.
Fast-moving stars can have origins other than a black hole, of course.
Ejection from the core curing is one plausible mechanism for producing such stars (Drukieretal.1999).. but their velocity vectors will be radial unlike a star in orbit around a black hole.
Ejection from the core during core-collapse is one plausible mechanism for producing such stars \citep{dcly}, but their velocity vectors will be radial unlike a star in orbit around a black hole.
We caution that the numbers presented. here are only estimates and depend on the scalines found by Cohu&IKulsrud (1978).
We caution that the numbers presented here are only estimates and depend on the scalings found by \citet{ck}. .
. Those models are sinele-mass anisotropic Planck simulations for the steady-state stellar distribution in the vicinity of the black hole.
Those models are single-mass anisotropic Fokker-Planck simulations for the steady-state stellar distribution in the vicinity of the black hole.
More modern models. which should include. at the very least. a range of stellar masses and a sell-consistent potential. will be needed to fully assess the significance of anv [ast-moving stars which are observed in these clusters.
More modern models, which should include, at the very least, a range of stellar masses and a self-consistent potential, will be needed to fully assess the significance of any fast-moving stars which are observed in these clusters.
The estimates made here also depend on the current central velocity clispersions in the elobular clusters.
The estimates made here also depend on the current central velocity dispersions in the globular clusters.
Since globular clusters can lose a large fraction of their mass due to stellar evolution and stellar-dvnanmical evolution. the numbers provided in Table 1 may well be underestimated if the proper velocity dispersion lo use in determining black hole masses is the original value. not the current one.
Since globular clusters can lose a large fraction of their mass due to stellar evolution and stellar-dynamical evolution, the numbers provided in Table \ref{T:top choices} may well be underestimated if the proper velocity dispersion to use in determining black hole masses is the original value, not the current one.
Further. the mass of any central black hole will have increased to some extent due to the capture of cluster stars.
Further, the mass of any central black hole will have increased to some extent due to the capture of cluster stars.
Using double the current velocity dispersion. for example. would increase NA. by a factor of 16 and VY. by a factor of 39. in which case the higher slope also predicts significant numbers of observable stars in the top few clusters.
Using double the current velocity dispersion, for example, would increase $N_{\rm M15}^L$ by a factor of 16 and $N_{\rm M15}^H $ by a factor of 39, in which case the higher slope also predicts significant numbers of observable stars in the top few clusters.
Given (he sensitivity (o these effects. obtaining reliable estimates lor the numbers of high-velocity stars will require fully evolving models.
Given the sensitivity to these effects, obtaining reliable estimates for the numbers of high-velocity stars will require fully evolving models.
Even in the event (hat proper-motion studies uncover no [ast moving stars. such models will allow for firm upper limits on the mass of anv black hole.
Even in the event that proper-motion studies uncover no fast moving stars, such models will allow for firm upper limits on the mass of any black hole.
C'ónstructing such models. and carrving out the necessary proper motion observations. is no small task. but given the strong interest and controversy currently surrounding (his topic. we believe that efforts along these lines should be vigorously pursued.
Constructing such models, and carrying out the necessary proper motion observations, is no small task, but given the strong interest and controversy currently surrounding this topic, we believe that efforts along these lines should be vigorously pursued.
This study was supported by a NASA LTSA grant NAG5-641404.
This study was supported by a NASA LTSA grant NAG5-6404.
of the jitter RAIS o, among all stars in the sample. where we term σῃ (hemagnitude of the Πίου. (
of the jitter RMS $\sigma_*$ among all stars in the sample, where we term $\sigma_0$ the of the jitter. (
In Section 5. we also trv an exponential distribution.)
In Section \ref{sec.tests} we also try an exponential distribution.)
The RMS jitter in an of stars with a Ravleigh distribution is v/2oy.
The RMS jitter in an of stars with a Rayleigh distribution is $\sqrt{2}\sigma_0$.
Our 6 hr systematic noise level of 3 can be explained if ση=1.8!.
Our 6 hr systematic noise level of 3 can be explained if $\sigma_0 = 1.8$.
. The predicted jitter distribution is best described by oy=1.72:0.1 (Figure 3)). consistent with our observations of 3. total RMS.
The predicted jitter distribution is best described by $\sigma_0 = 1.7\pm0.1$ (Figure \ref{fig.activity}) ), consistent with our observations of 3 total RMS.
Additional noise due to stellar rotation and starspots may occur on longer timescales (Barnesetal.2011).. ancl we perform caleulations with cy over the range 1.5-4.5!.
Additional noise due to stellar rotation and starspots may occur on longer timescales \citep{Barnes2011}, and we perform calculations with $\sigma_0$ over the range 1.5-4.5.
. However. we consider values near the upper limit. corresponding to an average svstematic noise of 6.5|. highly implausible because of the absence ofactive stus in in our sample (inset of Figure 3)).
However, we consider values near the upper limit, corresponding to an average systematic noise of 6.5, highly implausible because of the absence ofactive stars in in our sample (inset of Figure \ref{fig.activity}) ).
This is discussed further in Section 6..
This is discussed further in Section \ref{sec.discussion}.
We predict the distribution of RV. RAIS for each set of parameter values bv generating 10.000 Monte Carlo svstems. wilh host stars selected with replacement [rom the M2Ix survey. and orbital inclinations drawn from an isotropic distribution.
We predict the distribution of RV RMS for each set of parameter values by generating 10,000 Monte Carlo systems, with host stars selected with replacement from the M2K survey, and orbital inclinations drawn from an isotropic distribution.
EachKepler candidate planet has à probabilitv 1/5; of being added to each star.
Each candidate planet has a probability $1/s_i$ of being added to each star.
This ignores any autocorrelation between the presence of planets.
This ignores any autocorrelation between the presence of planets.
Masses are assigned to each planet using theAepler radius and (he MURR.
Masses are assigned to each planet using the radius and the MRR.
We ignore all planet candidates with radii larger than (he largest confirmed transiting planet (~2 Jupiter radii) as main sequence companions or false positives.
We ignore all planet candidates with radii larger than the largest confirmed transiting planet $\sim$ 2 Jupiter radii) as main sequence companions or false positives.
The RV variation induced by each planet is calculated from the planet mass. host star mass. and svstem inclination.
The RV variation induced by each planet is calculated from the planet mass, host star mass, and system inclination.
Orbits are assumed to be approximately coplanar (Lissaueretal.2011)..
Orbits are assumed to be approximately coplanar \citep{Lissauer2011b}.
Radial velocities are calculated using the actual epochs of observations and random mean anomalies al the first epoch.
Radial velocities are calculated using the actual epochs of observations and random mean anomalies at the first epoch.
We draw longitudes of perihelion from a uniform distribution ancl orbital eccentricities from a Ravleigh distribution with mean of 0.225 (Moorheadοἱal. 2011)..
We draw longitudes of perihelion from a uniform distribution and orbital eccentricities from a Rayleigh distribution with mean of 0.225 \citep{Moorhead2011}. .
We add formal and
We add formal and
To eusure that the lurge-scale diffuse cussion is nof due to the bendius of discrete sources. we present the total intensity radio contours at 1100 MIIz with the VLA in D configuration in the top panel of Fie. 3..
To ensure that the large-scale diffuse emission is not due to the blending of discrete sources, we present the total intensity radio contours at 1400 MHz with the VLA in D configuration in the top panel of Fig. \ref{radio_sub},
after subtraction of disete sources.
after subtraction of discrete sources.
To obtain it. we produced an inaege ofthe discrete sources by using only the lougest yvasclines of the D configuration data-set. and uuiforni weiehtins.
To obtain it, we produced an image of the discrete sources by using only the longest baselines of the D configuration data-set, and uniform weighting.