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\begin{document}

\title{Title}



\author[1]{John Phillips}%
\author[1]{Claudia}%
\affil[1]{University of Minnesota}%


\vspace{-1em}



  
  \date{June 12, 2024}


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Sample selection:

Dwarf galaxies - $8 < log M_{\star} 9$, $z < 0.02$, no AGN 

Massive galaxies - $10 < log M_{\star} < 11$, $z < 0.07$, no AGN

Comparison sample - Same except out to $z < 2 \times z_{max,sample}$. Incomplete, but we only use these objects to derive fiber corrections.

Dust correction - derive E(B-V) from Balmer decrement

Fiber corrections (requires involved discussion):

    -Define "overlap sample" between science sample and corresponding comparison sample:
        -"Sceince overlap sample" - objects in the science sample that do not fit into fiber at sample redshifts, but would at comparison redshifts

        -"Comparison overlap sample" - objects in the science sample that do fit into fiber at comparison redshifts, but would not at sample redshifts

        -At fixed size, how do we scale the science overlap sample so as to bring it into agreement with the comparison overlap sample?

        -Assume correction factor goes as $\Psi^{-\alpha}$, where $\Psi$ is the fraction of the galaxy's area contained within the fiber

        -Fit for alpha such that the median parameters (e.g. $H \alpha$ EW) of the overlap sample are equal\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Balmer-decrement1/Balmer-decrement1}
\caption{{Balmer decrement, from which we derive reddening corrections.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mass-size/mass-size}
\caption{{r band size mass relation, from which we derive galaxy sizes.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fiber-corrections/fiber-corrections}
\caption{{Cartoon depicting fiber correction process. By observing similar galaxies at lower and higher redshift, we can correct for the section of galaxy that falls outside the fiber.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/dwarf-8.0-EW-r/dwarf-8.0-EW-r}
\caption{{Replace this text with your caption%
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\includegraphics[width=0.70\columnwidth]{figures/dwarf-8.5-EW-r/dwarf-8.5-EW-r}
\caption{{Replace this text with your caption%
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\includegraphics[width=0.70\columnwidth]{figures/dwarf-9-EW-r/dwarf-9-EW-r}
\caption{{Replace this text with your caption%
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\includegraphics[width=0.70\columnwidth]{figures/dwarf-9.5-EW-r/dwarf-9.5-EW-r}
\caption{{Replace this text with your caption%
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\end{figure}\selectlanguage{english}
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\includegraphics[width=0.70\columnwidth]{figures/dwarf-10-EW-r/dwarf-10-EW-r}
\caption{{Replace this text with your caption%
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\end{figure}

Results:

- Plot SF ($H \alpha$ EW, luminosity, $H\alpha$/UV ratio) vs. size

    - anticorrelation seen at $10^9 M_{\odot}$, but not at $10^8 M_{\odot}$ or $10^{10} M_{\odot}$

    - trend expected under radial transport scenario

    - compare to simulations (FIRE) 

    - Is the lack of trend at $10^8 M_{\odot}$ due to incompleteness?

        - Important for core/cusp problem.

        - Local galaxies with cores are smaller than $10^9 M_{\odot}$
    
- What can fiber corrections tell us about radial distribution of star formation?

    - fiber differentials can determine average radial gradient in various properties

    - does logic make sense?


- If not radial transport, then what?

    - IMF?
    
    - Escape fraction?
    
    - Something else?

Use ''fiber differentials" to investigate radial distributions of parameters:

Nearby galaxies are missing some portion of a spectroscopic parameter $\rm Theta$, and some portion of their area due to area outside the fiber.

Define $\Psi$ such that $\rm A = \Psi A_{fiber}$. Furthermore, assume that $\rm \Theta = \Theta_{fiber} \Psi^{\alpha}$, where $\alpha$ is related to the radial dependence of $\Theta$.

To find out how $\alpha$ relates to theta, assume that $\Theta$ is a power law in $r$: $ \Theta(r) = \Theta_{fiber}  r^{\gamma}$. For a galaxy of radius R, we have:

$$\rm \Theta = 2\pi \int_{0}^{R} \Theta_{fiber} r^{\gamma+1} dr$$ 

$$\Theta = \frac{2 \pi \Theta_0}{\gamma+2} R^{\gamma+2}$$

From the definition of $\Psi$, we have $\rm \Psi = \frac{\pi R^2 }{ A_{fiber}}$. Therefore our assumption that we can correct for the $\Theta$ missing from the fiber via $\Psi^{\alpha}$ yields $\rm \Theta = \Theta_{fiber} \frac{\pi^\alpha R^{2\alpha} }{ A_{fiber}^{\alpha}}$.

Combining yields $R^{\gamma+2} \propto R^{2\alpha}$, i.e. $\gamma = 2\alpha -2$. We determine $\alpha$ in doing fiber corrections, giving us $\gamma.$\selectlanguage{english}
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\includegraphics[width=0.70\columnwidth]{figures/gradients/gradients}
\caption{{Replace this text with your caption%
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\end{figure}

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