Astronomy & Astrophysics

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$\begin{figure} \par\includegraphics[width=8cm,clip]{1016fig1.eps} \end{figure}$ Figure 1: ${\hat{x}}_y$ as a function of $\Delta$ for different values of $\epsilon$ according to Eq. (11): for $\epsilon =10^{-5}$ (solid), $\epsilon =10^{-4}$ (dashed), $\epsilon =10^{-3}$ (dotted) and $\epsilon =10^{-2}$ (dashed-dotted). The right ordinate corresponds to $\nu_y=k~ T_0\cdot{\hat{x}}_y/h$ , with T₀=2.725 K.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{1016fig2.eps} \end{figure}$ Figure 2: Frequency dependence of $g_i/g_1\equiv\Delta T_i/T_{\rm RJ}~\delta^i$ : Dotted line $\Delta T_1/T_{{\rm RJ}}~\delta$ , which is equivalent to a temperature distortion, solid line $g_y({\hat{x}})=2\Delta T_2/T_{{\rm RJ}}~\delta^2$ , which is equivalent to a y-distortion $\Delta T/y~T_0$ as given by Eq. (5), dashed line $\Delta T_3/T_{{\rm RJ}}~\delta^3$ and dashed-dotted line $\Delta T_4/T_{{\rm RJ}}~\delta^4$ . The function $g_y({\hat{x}})$ becomes larger than unity for $\nu\geq 146.2~$ GHz or ${\hat{x}}\geq 2.57$ , larger than 5 for $\nu\geq 396.7~$ GHz or ${\hat{x}}\geq 6.99$ and larger than 10 for $\nu\geq 681.3~$ GHz or ${\hat{x}}\geq 12$ . For $\nu = 100$ GHz or ${\hat{x}}= 1.76$ it follows g_y= 0.5. Also shown as vertical lines are the PLANCK LFI (dashed) and HFI (dash-dotted) frequency channels.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{1016fig3.eps} \end{figure}$ Figure 3: $\hat{x}_{\rm c}$ as a function of $\Delta$ for different values of $\epsilon$ according to Eq. (22) assuming that $\left <\delta ^k\right >=\Delta ^k$ : For $\epsilon =10^{-4}$ (solid), $\epsilon =10^{-3}$ (dashed) and $\epsilon =10^{-2}$ (dotted). The right ordinate corresponds to $\nu_{\rm c}=k~T_0\cdot\hat{x}_{\rm c}/h$ , with T₀=2.725 K.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig4.eps} \end{figure}$ Figure 4: Dependence of the inferred y-parameter on the chosen reference temperature $T_{\rm ref}$ for two blackbodies with temperatures T₁=2.72162 K and T₂=2.72838 K, corresponding to the minimum and maximum of the CMB dipole. In this case the mean RJ temperature is $T_{{\rm RJ}}=(T_1+T_2)/2=2.725~$ K and $y_{\rm b}=7.7\times 10^{-7}$ . - Sum of two blackbodies: total y-parameter as given by Eq. (38b) (dashed line). - Difference of two blackbodies: the absolute value $\vert y_{\rm dif}\vert$ of the inferred y-parameter as given in Eq. (41b) (solid line). The dotted vertical lines show the location of $T_{\rm ref}=T_1$ and $T_{\rm ref}=T_2$ respectively.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig5.ps} \end{figure}$ Figure 5: Angular distribution of $y_{\rm r}$ as given by Eq. (52) for $\theta _{\rm r}=20^\circ$ . The maximal spectral distortion is expected to appear in a ring perpendicular to the dipole axis and in this case has a value $y_{\rm r, max}=2.3\times 10^{-8}$ . The minimal y-parameter has a value of $y=2.4\times 10^{-10}$ and is located around the maximum and minimum of the dipole. The galactic plane was cut out using the kp0-mask of the WMAP data base.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1016fig6.ps} \end{figure}$ Figure 6: Dependence of the y-parameter in units of $y_{\rm d}=2.6~\times~ 10^{-7}$ on the beam radius $\theta _{\rm r}$ in degree for a circular beam: $y_{\rm r}$ for $\mu _0=\pm 1$ as given by Eq. (52) (dash-dot-dotted), $y_{\rm r, max}$ according to Eq. (53) (solid). For values $-1<\mu _0<1$ the curves lie between these two extremes.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig7.ps} \end{figure}$ Figure 7: Top: probability density p(y) for different aperture radii: $\theta _{\rm r}=0.25^\circ$ (solid), $\theta _{\rm r}=0.5^\circ$ (dashed) and $\theta _{\rm r}=1.0^\circ$ (dashed-dotted). - Bottom: dependence of the average y-parameter on the aperture radius $\theta _{\rm r}$ in degrees: for the beam spectral distortion (solid) and for the y-parameter resulting from the dispersion of the beam RJ temperatures in different directions on the sky relative to T₀ (dashed). Also shown is the maximal beam spectral distortion arising due to the dipole, $y_{\rm r, max}$ (dotted), according to Eq. (53).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig8.ps} \end{figure}$ Figure 8: Top: cumulative probability $p(y\geq y_0)$ for different aperture radii $\theta _{\rm r}=0.1^\circ$ (solid), $\theta _{\rm r}=0.25^\circ$ (dashed) and $\theta _{\rm r}=1.0^\circ$ (dashed-dotted). Bottom: dependence of the cumulative probability $p(y\geq y_0)$ on the aperture $\theta _{\rm r}$ for $y_0=8\times 10^{-10}$ (solid), $y_0=9\times 10^{-10}$ (dashed) and y₀=10^-9 (dashed-dotted).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig9.ps} \end{figure}$ Figure 9: Number of regions on the sky with $y\geq y_0$ for different aperture radii $\theta _{\rm r}$ in degree: for $y_0=8\times 10^{-10}$ (solid), $y_0=9\times 10^{-10}$ (dashed) and y₀=10^-9 (dashed-dotted).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fg10.eps} \end{figure}$ Figure 10: Illustration of a differential observing strategy: the maximum of the dipole lies in the direction of the z-axis. Both observed patches have the same radius $\theta _{\rm r}$ and the observed intensity difference is $\Delta I=I_2-I_1$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1016fg11.ps} \end{figure}$ Figure 11: Dependence of the y-parameter in units of $y_{\rm d}=2.6\times 10^{-7}$ on the beam radius $\theta _{\rm r}$ in degree for differential measurements: $y_{\mathcal{A},0,\rm opt}$ as given by Eq. (73) (dashed/left ordinate), $y_{\mathcal{A},1,\rm opt}$ as given by Eq. (76) (solid/right ordinate) and $y_{\mathcal{A},2,\rm opt}$ as given by Eq. (77) (dotted/left ordinate).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{1016fig1.eps} \end{figure}$	Figure 1: ${\hat{x}}_y$ as a function of $\Delta$ for different values of $\epsilon$ according to Eq. (11): for $\epsilon =10^{-5}$ (solid), $\epsilon =10^{-4}$ (dashed), $\epsilon =10^{-3}$ (dotted) and $\epsilon =10^{-2}$ (dashed-dotted). The right ordinate corresponds to $\nu_y=k~ T_0\cdot{\hat{x}}_y/h$ , with T₀=2.725 K.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{1016fig3.eps} \end{figure}$	Figure 3: $\hat{x}_{\rm c}$ as a function of $\Delta$ for different values of $\epsilon$ according to Eq. (22) assuming that $\left <\delta ^k\right >=\Delta ^k$ : For $\epsilon =10^{-4}$ (solid), $\epsilon =10^{-3}$ (dashed) and $\epsilon =10^{-2}$ (dotted). The right ordinate corresponds to $\nu_{\rm c}=k~T_0\cdot\hat{x}_{\rm c}/h$ , with T₀=2.725 K.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1016fig6.ps} \end{figure}$	Figure 6: Dependence of the y-parameter in units of $y_{\rm d}=2.6~\times~ 10^{-7}$ on the beam radius $\theta _{\rm r}$ in degree for a circular beam: $y_{\rm r}$ for $\mu _0=\pm 1$ as given by Eq. (52) (dash-dot-dotted), $y_{\rm r, max}$ according to Eq. (53) (solid). For values $-1<\mu _0<1$ the curves lie between these two extremes.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fig9.ps} \end{figure}$	Figure 9: Number of regions on the sky with $y\geq y_0$ for different aperture radii $\theta _{\rm r}$ in degree: for $y_0=8\times 10^{-10}$ (solid), $y_0=9\times 10^{-10}$ (dashed) and y₀=10^-9 (dashed-dotted).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1016fg10.eps} \end{figure}$	Figure 10: Illustration of a differential observing strategy: the maximum of the dipole lies in the direction of the z-axis. Both observed patches have the same radius $\theta _{\rm r}$ and the observed intensity difference is $\Delta I=I_2-I_1$ .
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{1016fg11.ps} \end{figure}$	Figure 11: Dependence of the y-parameter in units of $y_{\rm d}=2.6\times 10^{-7}$ on the beam radius $\theta _{\rm r}$ in degree for differential measurements: $y_{\mathcal{A},0,\rm opt}$ as given by Eq. (73) (dashed/left ordinate), $y_{\mathcal{A},1,\rm opt}$ as given by Eq. (76) (solid/right ordinate) and $y_{\mathcal{A},2,\rm opt}$ as given by Eq. (77) (dotted/left ordinate).
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