# Satellite Galaxy Velocity Dispersions in the SDSS and Modified Gravity Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data Analysis

**Figure 1.**Line-of-sight velocities (

**a**) and radial number densities (

**b**) of candidate satellite galaxies (full sample) as a function of projected distance from the candidate host. After removal of candidate interlopers, the radial number density (

**c**) follows a power law profile with an exponent of $-1.5$ (

**d**).

## 3. Satellite Galaxy Velocity Dispersion

^{2}. The form of the function $\mu \left(x\right)$ originally proposed by Milgrom [5] is given by $\mu \left(x\right)=x/\sqrt{1+{x}^{2}}$; however, better fits and better asymptotic behavior are achieved using $\mu \left(x\right)=x/(1+x)$ [1].

## 4. The Interloper Background

**Figure 2.**Likelihood of ΛCDM, Modified Newtonian Dynamics (MOND), Modified Gravity Theory (MOG), and the Newtonian model without exotic dark matter, as a function of the κ parameter as defined in Equation (20) (

**top**: $-20.5>{M}_{g}^{\left(1\right)}>-21.1$,

**bottom**: $-21.1>{M}_{g}^{\left(2\right)}>-21.6$). Horizontal lines indicate the 1σ and 2σ levels relative to the maximum likelihood of the best performing model (ΛCDM).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Appendix

## Data Analysis

SELECT G.ra,G.dec,u,g,r,i,dered_z, petroMag_r, psfMag_r,extinction_r,petror50_r,S.z,S.zErr, S.zConf,G.flags,S.zStatus FROM Galaxy as G, SpecObj as S WHERE G.ObjID = S.BestObjID

- right ascension (α),
- declination (δ),
- u-band extinction-corrected model magnitude (m
_{u}), - g-band extinction-corrected model magnitude (m
_{g}), - r-band extinction-corrected model magnitude (m
_{r}), - i-band extinction-corrected model magnitude (m
_{i}), - z-band extinction-corrected model magnitude (m
_{z}), - Petrosian r-band magnitude m
_{P}(needed for LRG selection), - PSF r-band magnitude m
_{PSF}(needed for LRG selection), - r-band extinction e
_{r}, - Petrosian r-band 50% light radius r
_{50}, - redshift (z),
- redshift error (Δz),
- redshift correlation coefficient,
- photometric flags (as defined by the SDSS),
- redshift status (as defined by the SDSS).

## Additional Details

**Table A1.**Polynomial coefficients used to approximate Equation (6).

Coefficient | Model | ||
---|---|---|---|

Newton | MOND | MOG | |

${K}_{0}=$ | –393.5231055300000 | –395.607713960000 | –393.443209400000 |

${K}_{1}=$ | –196.0740660300000 | –187.363184940000 | –190.049179440000 |

${K}_{2}=$ | –047.9673574490000 | –037.987866538000 | –042.953121084000 |

${K}_{3}=$ | 00–7.3231807355000 | –004.376376109200 | 00–7.099396442600 |

${K}_{4}=$ | –000.7122426667800 | 00–2.992885212100 | –002.445117923400 |

${K}_{5}=$ | 00–0.0402686798220 | –000.443019466890 | 00–0.491987549850 |

${K}_{6}=$ | –000.0010225183241 | 00–0.021854265856 | –000.031686326614 |

**Table A2.**Normalization factors for Equation (18).

Model | ${\alpha}_{S}$ |
---|---|

Newton | 0.810 |

MOND | 0.880 |

MOG | 0.849 |

ΛCDM | 0.572 |

## Conflicts of Interest

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Moffat, J.W.; Toth, V.T. Satellite Galaxy Velocity Dispersions in the SDSS and Modified Gravity Models. *Galaxies* **2014**, *2*, 263-274.
https://doi.org/10.3390/galaxies2020263

**AMA Style**

Moffat JW, Toth VT. Satellite Galaxy Velocity Dispersions in the SDSS and Modified Gravity Models. *Galaxies*. 2014; 2(2):263-274.
https://doi.org/10.3390/galaxies2020263

**Chicago/Turabian Style**

Moffat, John W., and Viktor T. Toth. 2014. "Satellite Galaxy Velocity Dispersions in the SDSS and Modified Gravity Models" *Galaxies* 2, no. 2: 263-274.
https://doi.org/10.3390/galaxies2020263