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In the second case, the changes were almost indiscernable. | In the second case, the changes were almost indiscernable. |
The global energy varies slightly as the change in Κα. | The global energy varies slightly as the change in $R_\alpha$. |
In order to demonstrate the roles of the parameters Περ and Binj, we show in Fig. | In order to demonstrate the roles of the parameters $n_{\rm sp}$ and $B_{\rm inj}$, we show in Fig. |
8 the field configurations at t=17 (corresponding to an age of approximately GGyr) for models with Ry=1,R,10 and Bi,=1,2,4 (i.e., increasing Bj, from the value used in the models describedin Sect. 3.1)). | \ref{fig:nspoteq50} the field configurations at $t=17$ (corresponding to an age of approximately Gyr) for models with $R_\alpha=1, R_\omega=10$ and $B_{\rm
inj}=1, 2, 4$ (i.e., increasing $B_{\rm inj}$ from the value used in the models describedin Sect. \ref{changeromega}) ). |
Reducing the value of Περ to 50 reduces the amount of disorder in the field (compare the last panel of Fig. | Reducing the value of $n_{\rm sp}$ to 50 reduces the amount of disorder in the field (compare the last panel of Fig. |
3 with panel (a) of Fig. 8)). | \ref{fig:model138} with panel (a) of Fig. \ref{fig:nspoteq50}) ). |
As less small-scale field is being injected, this is perhaps not so surprising. | As less small-scale field is being injected, this is perhaps not so surprising. |
We also now consider cases in which the turbulence in the star-forming regions (spots) is assumed to be significantly stronger than in the general ISM, giving an injected field larger than the general equipartition field, i.e. Bijj>1. | We also now consider cases in which the turbulence in the star-forming regions (spots) is assumed to be significantly stronger than in the general ISM, giving an injected field larger than the general equipartition field, i.e. $B_{\rm inj}>1$. |
Note that we do not change the alpha-quenching formalism or the value of Βιπ] in the spots when Bij>1, as the dominant dynamo mechanism in the spots is assumed to be a small-scale dynamo to which we cannot apply the alpha-quenching concept. | Note that we do not change the alpha-quenching formalism or the value of $B_{\rm
inj}$ in the spots when $B_{\rm inj}>1$, as the dominant dynamo mechanism in the spots is assumed to be a small-scale dynamo to which we cannot apply the alpha-quenching concept. |
Moreover the filling factor of the high field region of the spots is small, and the usual aw dynamo operates through nearly all the disc. | Moreover the filling factor of the high field region of the spots is small, and the usual $\alpha\omega$ dynamo operates through nearly all the disc. |
These assumptions are strongly supported by an experiment (not shown) where with the parameters of panel (c) of Fig. 8,, | These assumptions are strongly supported by an experiment (not shown) where with the parameters of panel (c) of Fig. \ref{fig:nspoteq50}, |
a alpha-quenching «=αο/(1+E), with E a measure of the mean magnetic energy over the disc, was used. | a alpha-quenching $\alpha=\alpha_0/(1+E)$, with $E$ a measure of the mean magnetic energy over the disc, was used. |
The resulting field vectors were very similar to those shown in panel (c) of Fig. 8.. | The resulting field vectors were very similar to those shown in panel (c) of Fig. \ref{fig:nspoteq50}. |
As Bin; increases, obviously does the disorder: compare panels (a), (b), (c) of Fig. 8.. | As $B_{\rm inj}$ increases, obviously does the disorder: compare panels (a), (b), (c) of Fig. \ref{fig:nspoteq50}. |
Further, the reduction of Περ seems to produce some overall change to the field structure. | Further, the reduction of $n_{\rm sp}$ seems to produce some overall change to the field structure. |
Specifically, the reduced value of Περ means, quite predictably, that for given By; the field is less disordered. | Specifically, the reduced value of $n_{\rm sp}$ means, quite predictably, that for given $B_{\rm inj}$ the field is less disordered. |
Less predictably, perhaps, even when Bj is increased and more disorder is apparent, the spiral structure is better defined and rathermore open, until Bi,= 4, see the successive panels of Fig. 8.. | Less predictably, perhaps, even when $B_{\rm inj}$ is increased and more disorder is apparent, the spiral structure is better defined and rathermore open, until $B_{\rm inj}=4$ , see the successive panels of Fig. \ref{fig:nspoteq50}. . |
'To come to a resolution, we first obtain some results on conditional measurement probabilities. | To come to a resolution, we first obtain some results on conditional measurement probabilities. |
Say we have two entangled (but spatially separated) particles A and B, and we want to measure observables O4 and Og. | Say we have two entangled (but spatially separated) particles $A$ and $B$, and we want to measure observables $\mathcal{O}_A$ and $\mathcal{O}_B$. |
If we have corresponding eigenbases |?)4 and |j)p, we can write a general state as ΑΙ) B(2) with ο=1. | If we have corresponding eigenbases $|i\rangle _A $ and $|j\rangle _B $, we can write a general state as _A _B with $\sum_{i j} |\alpha_{ij}|^2=1$. |
For simplicity, we assume that the above states all have the same energy. | For simplicity, we assume that the above states all have the same energy. |
Time evolution then involves an irrelevant overall phase factor, which we suppress. | Time evolution then involves an irrelevant overall phase factor, which we suppress. |
When measuring O4, the chance that the outcome O4 equals the I-th eigenvalue a; is given by |. | When measuring $\mathcal{O}_A$ the chance that the outcome $O_A$ equals the $I$ -th eigenvalue $a_I$ is given by |^2. |
And similar for P(Og=b;). | And similar for $P(O_B=b_J)$. |
Say we have indeed measured on B and got Og=by. | Say we have indeed measured on $B$ and got $O_B=b_J$. |
The state then collapses onto and for this |) the chance to measure O4=a; is P(O4=ar|OBby) In words: measuring Og has resulted in a change in probabilities (3)) + @)). | The state then collapses onto and for this $|\psi'\rangle $ the chance to measure $O_A=a_I$ is P(O_A=a_I|O_B=b_J)= In words: measuring $\mathcal{O}_B$ has resulted in a change in probabilities \ref{eq:original prob}) ) $\rightarrow$ \ref{eq:new prob}) ). |
So the combined chance to first find Og=b; and then OA=αι is given by P(O4=a7|Op67) P(Op=b;)(6) Now what is the chance for this to happen in the other order? | So the combined chance to first find $O_B=b_J$ and then $O_A=a_I$ is given by P(O_A=a_I|O_B=b_J) P(O_B=b_J) Now what is the chance for this to happen in the other order? |
That is, what is the chance to measure ar, and then find Og=bj? | That is, what is the chance to measure $O_A=a_I$ , and then find $O_B=b_J$? |
This is given by P(Op=b;|O4 aj)P(O4=ar)(7) And the last two expressions are equal by Bayes' theorem. ( | This is given by P(O_B=b_J|O_A=a_I) P(O_A=a_I) And the last two expressions are equal by Bayes' theorem. ( |
Both are of course just equal to |orr; |?.) | Both are of course just equal to $|\alpha_{IJ}|^2$ .) |
None of this looks very surprising, but we want to stress that the total probability to find O4=a; and Op=b; at which the measurements occur. | None of this looks very surprising, but we want to stress that the total probability to find $O_A=a_I$ and $O_B=b_J$ at which the measurements occur. |
Before we interpret this simple but important result, we generalize it a to performing measurements on the two particles. | Before we interpret this simple but important result, we generalize it a to performing measurements on the two particles. |
So consider observables O4,1:::O4,, and Ogi:::OBm- | So consider observables $\mathcal{O}_{A,1}\cdots\,\mathcal{O}_{A,n}\,\,$ and $\,\,\mathcal{O}_{B,1}\cdots\,\mathcal{O}_{B,m}\,$. |
We still suppose A and B are each in an energy eigenstate, so that time evolution is trivial. | We still suppose $A$ and $B$ are each in an energy eigenstate, so that time evolution is trivial. |
Now consider for each operator specific eigenvalue. | Now consider for each operator specific eigenvalue. |
We denote these by A? and AP. | We denote these by $\lambda^A_i$ and $\lambda^B_j$. |
If we start with a joint state |v) and measure O4,:-:O4,, and Opi-::Ομ (in that order) then the chance to obtain as outcomes the numbers (A---AZ,AP...AB} is given by lambda,lambda”, Where Έλα and Ds are the projectors on the eigenspaces corresponding to the eigenvalues A# and AP⋅ | If we start with a joint state $|\psi\rangle $ and measure $\mathcal{O}_{A,1}\cdots\,\mathcal{O}_{A,n}\,\,$ and $\,\,\mathcal{O}_{B,1}\cdots\,\mathcal{O}_{B,m}\,$ (in that order) then the chance to obtain as outcomes the numbers $\{\lambda^A_1\cdots\lambda^A_n,\lambda^B_1\cdots\lambda^B_m\}$ is given by ^B_m) = Where $P_{\lambda^A_i}$ and $P_{\lambda^B_j}$ are the projectors on the eigenspaces corresponding to the eigenvalues $\lambda^A_i$ and $\lambda^B_j$. |
Now the measurements on each particle don’t have to be compatible, so the [Pya,Ελλ are not necessarily zero, and similar for B. | Now the measurements on each particle don't have to be compatible, so the $[P_{\lambda^A_i},P_{\lambda^A_j}]$ are not necessarily zero, and similar for $B$. |
However, the product structure of the Hilbert space implies [Pya,ΕλΡ]=0 for all { and 1. | However, the product structure of the Hilbert space implies $[P_{\lambda^A_i},P_{\lambda^B_j}]=0$ for all $i$ and $j$. |
So in the right hand side, we can freely move the P4s through the P4, as long as we preserve the order of the Pys and the Py4 amongst themselves. | So in the right hand side, we can freely move the $P_{\lambda^B}$ through the $P_{\lambda^A}$, as long as we preserve the order of the $P_{\lambda^B}$ and the $P_{\lambda^A}$ amongst themselves. |
For example, we can drag Py» completely to the left. | For example, we can drag $P_{\lambda^B_1}$ completely to the left. |
This implies that -tambdaB)= tona tomasΒ){9 In words: the probability of a collection of outcomes does not depend on the relative moments of measurements on A and B. | This implies that ^A_n ^B_m) = ^A_n ^B_m) In words: the probability of a collection of outcomes does not depend on the relative moments of measurements on $A$ and $B$. |
As long as we respect the order of measurements on each particle separately, the chance stays the same. | As long as we respect the order of measurements on each particle separately, the chance stays the same. |
This generalizes the previous Although the above expressions are all very simple, the result is, upon second thought, very non-trivial. | This generalizes the previous Although the above expressions are all very simple, the result is, upon second thought, very non-trivial. |
It shows that in general, the relative time ordering of measurements on separated (but possible entangled) particles A and D doesn't matter at all. ( | It shows that in general, the relative time ordering of measurements on separated (but possible entangled) particles $A$ and $B$ doesn't matter at all. ( |
Of course,for measurements on the same particle, the order matter. | Of course,for measurements on the same particle, the order matter.) |
What does this say about the collapse of the total wave function? | What does this say about the collapse of the total wave function? |
To see this, it might be elusive to write the projection operators from the previous section as their action on the product Hilbert space H4&Hg of | To see this, it might be elusive to write the projection operators from the previous section as their action on the product Hilbert space $\mathcal{H}_A\otimes \mathcal{H}_B$ of |
the apparent pressure of (he LB?” | the apparent pressure of the LB?” |
To gain insights on this question. one can observe stars located behind individual clouds inside the LB and measure (heir absorption features of C I arising from different. fine-structure levels of the ground state. | To gain insights on this question, one can observe stars located behind individual clouds inside the LB and measure their absorption features of C I arising from different fine-structure levels of the ground state. |
The ratios of populations of (hese states are governed by an equilibrium between collisional excitations (governed by local densities ancl temperatures) ancl spontaneous radiative decays. | The ratios of populations of these states are governed by an equilibrium between collisional excitations (governed by local densities and temperatures) and spontaneous radiative decays. |
Stars sultable for viewing the C I features had to satisfy a number of conditions to vield useful results. | Stars suitable for viewing the C I features had to satisfy a number of conditions to yield useful results. |
Their selection is described in relplan:: ultimately. four such stars were observed in a manner described in relobs.. | Their selection is described in \\ref{plan}; ultimately four such stars were observed in a manner described in \\ref{obs}. |
The € I absorption features are very weak. and in order to obtain useful measures of their strengths particular care was exercised (o remove spurious signals arising [from the detector. as outlined in relartifacts.. ( | The C I absorption features are very weak, and in order to obtain useful measures of their strengths particular care was exercised to remove spurious signals arising from the detector, as outlined in \\ref{artifacts}. ( |
Mathematical details about the correction method are presented separately in the Appendix.) | Mathematical details about the correction method are presented separately in the Appendix.) |
Section ?? deseribes how the equivalent widths of various absorption features were combined ancl corrected for saturation (very mild. except for one of the stars). | Section \ref{analysis} describes how the equivalent widths of various absorption features were combined and corrected for saturation (very mild, except for one of the stars). |
This section also discusses the derivations of fine-structure population ratios. which max be compared to the theoretically expected values for different conditions relexpected;1)). | This section also discusses the derivations of fine-structure population ratios, which may be compared to the theoretically expected values for different conditions \\ref{expected_f1}) ). |
Be foreonecanderiveusefullimilsforthelhermalpressures.theallowablerangesoflHemper reflempimils. | Before one can derive useful limits for the thermal pressures, the allowable ranges of temperature must be constrained, and different methods of deriving these constraints are discussed in \\ref{temp_limits}. |
Ullimalely.thelimilsforthepopulationratiosandlemperaluresrestricllhepossiblevaluesf summe | Ultimately, the limits for the population ratios and temperatures restrict the possible values for $p/k$ [and local density $n({\rm H})$ ], as shown for the four cases in the diagrams presented in Fig. \ref{limit_panels}. |
ary. | Three out of the four stars indicate internal thermal pressures for the foreground clouds that are below the generally accepted value for the LB. |
. | Possible explanations for this imbalance, duplicating that seen for LIC, are presented in \\ref{summary}. |
Target stars chosen for the survey had to satisfy [our fundamental criteria. | Target stars chosen for the survey had to satisfy four fundamental criteria. |
First. the stars had to be within the Local Bubble or near its edge. | First, the stars had to be within the Local Bubble or near its edge. |
Second. thev had to have sufficient neulral eas in front so that there was a reasonable expectation of seeing the C I absorption features. | Second, they had to have sufficient neutral gas in front so that there was a reasonable expectation of seeing the C I absorption features. |
Third. the stars had to be bright enough to give a good signal-to-noise ratio in a reasonably short observing time. | Third, the stars had to be bright enough to give a good signal-to-noise ratio in a reasonably short observing time. |
Finally. (he survey avoided stars with projected rotational velocities esin;<50kms! (Uesugi Fukuda 1981 — with actual values provided bv the VizieR web site at the Strasbourg Data Center). so that stellar. features would. not cause confusion when the interstellar lines were being identified and measured. | Finally, the survey avoided stars with projected rotational velocities $v\sin i <
50\,{\rm km~s}^{-1}$ (Uesugi Fukuda 1981 – with actual values provided by the VizieR web site at the Strasbourg Data Center), so that stellar features would not cause confusion when the interstellar lines were being identified and measured. |
To satisfy the first two requirements. the selection included stars less than about. LOO pe away that had interstellar D-line absorption features indicating 101 N(Na D <3x10!em7. | To satisfy the first two requirements, the selection included stars less than about 100 pc away that had interstellar D-line absorption features indicating $10^{11}<N$ (Na I) $< 3\times 10^{11}\,{\rm cm}^{-2}$. |
A compilation by Welsh et al (1994) was a good source of inlormation about D-line absorption al (he time the survey was being planned. | A compilation by Welsh et al (1994) was a good source of information about D-line absorption at the time the survey was being planned. |
Fhixes at | Fluxes at |
considering the possible dependence of planet formation on composition. Ml] is à better index to use than Fe/1I] (Gonzalez2009). | considering the possible dependence of planet formation on composition, [M/H] is a better index to use than [Fe/H] \citep{gg09}. |
. Pherefore. we have replaced. Fe/H] with MH] when calculating the Ay index. | Therefore, we have replaced [Fe/H] with [M/H] when calculating the $\Delta_{\rm 1}$ index. |
We show in Figure la the bias-correctecd weighted average vsini dillerences between the SAWP and. non-SWP? samples (GNvsini). | We show in Figure 1a the bias-corrected weighted average vsini differences between the SWP and non-SWP samples $\Delta$ vsini). |
We corrected the Avsini values for bias in the same way as described in Gonzalezetal.(2010a). | We corrected the $\Delta$ vsini values for bias in the same way as described in \citet{gg10a}. |
. Driellv. the method involves splitting the non-SWID sample into two subsamples. | Briefly, the method involves splitting the non-SWP sample into two subsamples. |
We then caleulatecl Avsini values rom these subsamples in the same wav as was done with he original SWDP anc non-SWDI? samples. | We then calculated $\Delta$ vsini values from these subsamples in the same way as was done with the original SWP and non-SWP samples. |
Any trends in hese Avsini values with Tr are considered. biases. | Any trends in these $\Delta$ vsini values with $_{\rm eff}$ are considered biases. |
The results presented. in Figure la resemble those in Figure 12 of Gonzalezctal.(20102) which was also based on he data of ValentianclFischer(2005). | The results presented in Figure 1a resemble those in Figure 12 of \citet{gg10a}, , which was also based on the data of \citet{vf05}. |
.. As in Gonzalezοἱ (20102)... we subtracted the average linear (bias) trend rom the non-SWP? ;Nvsini- Tar data from the οΛΑΟ Avsini values. | As in \citet{gg10a}, we subtracted the average linear (bias) trend from the non-SWP $\Delta$ $_{\rm eff}$ data from the SWP $\Delta$ vsini values. |
However. as is evident in Figure 11 of Gonzalezetal. (2010a).. the required bias correction is not quite linear with Tr. | However, as is evident in Figure 11 of \citet{gg10a}, the required bias correction is not quite linear with $_{\rm eff}$. |
For this reason. we also corrected for bias using a second method. | For this reason, we also corrected for bias using a second method. |
We caleulated the vsini difference values in increments of 100 Ix using the non-SWP sample and applied these olfsets to the vsini dillerences between the ολλος and non-SWPs. | We calculated the vsini difference values in increments of 100 K using the non-SWP sample and applied these offsets to the vsini differences between the SWPs and non-SWPs. |
We show the results of this approach in Figure lb. | We show the results of this approach in Figure 1b. |
Phe overall pattern is similar to that in Figure la. but the distribution of vsini dillerences is Hlatter between 5500 and 6000 Ix. Phe average Avsini value for the data plotted in Figure Ib between these two temperatures is 0.46c0.96 (s.c.) | The overall pattern is similar to that in Figure 1a, but the distribution of vsini differences is flatter between 5500 and 6000 K. The average $\Delta$ vsini value for the data plotted in Figure 1b between these two temperatures is $-0.46 \pm 0.96$ (s.e.) |
dOL) (s.m) | $\pm~0.11$ (s.e.m.) |
km L. LE | km $^{\rm -1}$. |
SThe correspondingB average Avsini value determined by Gonzalezetal.(2010a) from the ValentiandFischer(2005) data is 0.66d1.08 (s.c.) | The corresponding average $\Delta$ vsini value determined by \citet{gg10a} from the \citet{vf05} data is $-0.66 \pm 1.08$ (s.e.) |
+£0.13 (sean) | $\pm~0.13$ (s.e.m.) |
km s.1 (for Tar=5550 to 6000 Ix). | km $^{\rm -1}$ (for $_{\rm eff} = 5550$ to 6000 K). |
ValentiandFischer(2005). reported the abundances of five elements: Na. Si. Ti. Fe and Ni. | \citet{vf05} reported the abundances of five elements: Na, Si, Ti, Fe and Ni. |
While this is a small number of elements for our analysis. their T. values span nearly the same range as the more extensive set of elements emploved bv Gonzalezetal.(20100b). | While this is a small number of elements for our analysis, their $_{\rm c}$ values span nearly the same range as the more extensive set of elements employed by \citet{gg10b}. |
. In. addition. the ValentiandFischer(2005). dataset is large and the abundances have small uncertainties. | In addition, the \citet{vf05} dataset is large and the abundances have small uncertainties. |
Phe samples of SWDPs ancl non-SWI's we use in this sections are the same ones we used for the vsini analysis above. | The samples of SWPs and non-SWPs we use in this sections are the same ones we used for the vsini analysis above. |
We calculated the abuncance-P. slope for cach star with simple linear least-squares. | We calculated the $_{\rm c}$ slope for each star with simple linear least-squares. |
We then calculated the weighted-average N/LH]-T. dillerences between the SWPs and non-SWDPs using the same procedure as described above. | We then calculated the weighted-average $_{\rm c}$ differences between the SWPs and non-SWPs using the same procedure as described above. |
However. in this case the bias corrections are small. allowing us to adjust the data with a simple linear fit. as in Gonzalez (2010b). | However, in this case the bias corrections are small, allowing us to adjust the data with a simple linear fit, as in \citet{gg10b}. |
. We show the corrected data in Figure 2. | We show the corrected data in Figure 2. |
lt resembles the data in Figure 2 of Gonzalezetal.(2010b). | It resembles the data in Figure 2 of \citet{gg10b}. |
. Gonzalezetal.(2010b) also confirmed. the discovery bv Melendezetal.(2000). that the more metal-rich SWDPs display more negative N/IIJ]-F.. slopes than non-SWPs.while more metal-poor ολλον don't. | \citet{gg10b} also confirmed the discovery by \citet{mel09} that the more metal-rich SWPs display more negative $_{\rm c}$ slopes than non-SWPs,while more metal-poor SWPs don't. |
We find. very. similar patterns in the current data. which we show in Figure 3. | We find very similar patterns in the current data, which we show in Figure 3. |
IsaacsonandFischer(2010). present chromospheric activity measurements of more than 2600 stars on the California Planet Search Program. | \citet{Isaac10} present chromospheric activity measurements of more than 2600 stars on the California Planet Search Program. |
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