This paper investigates the statistical properties of the Tully-Fisher relations for a volume limited complete sample of spiral galaxies in the nearby Ursa Major cluster .
The merits of B , R , I and K ^ { \prime } surface photometry and the availability of detailed kinematic information from HI synthesis imaging have been exploited .
In addition to the corrected HI global profile widths W _ { R,I } ^ { \mbox { \scriptsize i } } , the available HI rotation curves allow direct measurements of the observed maximum rotational velocities V _ { \mbox { \scriptsize max } } and the amplitudes V _ { \mbox { \scriptsize flat } } of the outer flat parts .
The dynamical state of the gas disks could also be determined in detail from the radio observations .
The four luminosity and three kinematic measures allowed the construction of twelve correlations for various subsamples .
For large galaxy samples , the M ^ { \mbox { \scriptsize b,i } } _ { \mbox { \scriptsize R } } - Log ( W _ { R,I } ^ { \mbox { \scriptsize i } } ) correlation in conjunction with strict selection criteria is preferred for distance determinations with a 7 % accuracy .
Galaxies with rotation curves that are still rising at the last measured point , lie systematically on the low velocity side of the TF-relation .
Galaxies with a partly declining rotation curve ( V _ { \mbox { \scriptsize max } } > V _ { \mbox { \scriptsize flat } } ) tend to lie systematically on the high velocity side of the relation when using W _ { R,I } ^ { \mbox { \scriptsize i } } or V _ { \mbox { \scriptsize max } } .
However , systematic offsets are eliminated when V _ { \mbox { \scriptsize flat } } is used .
Residuals of the M ^ { \mbox { \scriptsize b,i } } _ { \mbox { \scriptsize B } } - Log ( 2V _ { \mbox { \scriptsize flat } } ) relation correlate consistently with global galaxy properties along the Hubble sequence like morphological type , color , surface brightness and gas mass fraction .
These correlations are absent for the near-infrared M ^ { \mbox { \scriptsize b,i } } _ { \mbox { \scriptsize$K ^ { \prime } $ } } - Log ( 2V _ { \mbox { \scriptsize flat } } ) residuals .
The tightest correlation ( \chi ^ { 2 } _ { \mbox { \scriptsize red } } =1.1 ) is found for the M ^ { \mbox { \scriptsize b,i } } _ { \mbox { \scriptsize$K ^ { \prime } $ } } –Log ( 2V _ { \mbox { \scriptsize flat } } ) relation which has a slope of - 11.3 \pm 0.5 and a total observed scatter of 0.26 magnitudes with a most likely intrinsic scatter of zero .
The tightness of the near-infrared correlation is preserved when converting it into a baryonic TF-relation which has a slope of - 10.0 in case ( { \cal M } _ { \mbox { \scriptsize gas } } /L _ { \mbox { \scriptsize$K ^ { \prime } $ } } ) =1.6 while a zero intrinsic scatter remains most likely .
Based on the tightness of the near-infrared and baryonic correlations , it is concluded that the Tully-Fisher relation reflects a fundamental correlation between the mass of the dark matter halo , measured through its induced maximum rotational velocity V _ { \mbox { \scriptsize flat } } , and the total baryonic mass { \cal M } _ { \mbox { \scriptsize bar } } of a galaxy where { \cal M } _ { \mbox { \scriptsize bar } } \propto V ^ { 4 } _ { \mbox { \scriptsize flat } } .
Although the actual distribution of the baryonic matter inside halos of similar mass can vary significantly , it does not affect this relation .