1. Introduction Classical Cepheids are variable stars whose magnitudes oscillate with periods of 1 to 50 days. In the 1910s, Henrietta Swan Leavitt discovered the Period-Luminosity Relation, m = a + blogP, while observing Cepheids in the Large Magellanic Cloud. If we know the distance, d, to the LMC, we can find a P-L relation for absolute magnitude: M = a + blogP – 5log 10 (d / 10 pc) If we assume that Cepheids in a particular target galaxy obey the same P-L relation as those in the LMC, we can use the LMC as a calibrating galaxy. That is, we use a, b, and d of the LMC to ascertain the distance of the target galaxy. Thus far, the LMC has been the calibrating galaxy. Its distance modulus is known with an uncertainty of about 25% [1]. However, the uncertainty in distance modulus to NGC 4258 is expected to be reduced to under 3% [1]. Consequently, it has been suggested that NGC 4258 be used as a calibrating galaxy. This has prompted investigations into the nature of the P-L relation in NGC 4258. There is controversy over whether it is non-linear, with a break at P = 10 days, i.e. m = { a 1 + b 1 logP, P < 10; a 2 + b 2 logP, P ≥ 10}. Wesenheit magnitude is de-reddened: W = I – 1.55(V-I). We would also like to test the non-linearity of the P-W relation and the P-C relation. We investigate non-linearity of the P-L, P-C, and P-W relations in NGC 4258 using the F test. F Test The F test compares the fits of two curves to the same set of data, considering not only their residual sums of squares, but their degrees of freedom. For example, a curve with 3 degrees of freedom will have a lower F-statistic than a curve with 2 degrees of freedom that yields the same RSS. The F statistic is defined as follows: F = (RSS R – RSS F ) × (v R – v F ) RSS R = residual sum of squares of linear model RSS F v F RSS F = RSS of non-linear model v R = degrees of freedom of linear model (N – 2) v F = degrees of freedom of non-linear model (N – 4) N = number of observations We then use F, nuR, and nuF to find the probability of observing data with an F statistic greater than or equal to our F if the null (reduced) hypothesis is true. Non-Linearity of NGC 4258 All NGC 4258 data were taken from Macri et al [1] In all graphs, the blue line is the linear fit and the green lines are the non-linear fit. F statistic = 0.4114 Probability = 0.6633 Conclusion: Linear F statistic = 5.1446 Probability = 0.0066 Conclusion: Non-Linear F Statistic = 0.7998 Probability = 0.4508 Conclusion: Linear Conclusion It seems inconsistent that a linear P-L relation and non-linear P-C relation should yield a linear P-W relation. It seems likely that the P-C relation is more sensitive to the 10 day break than the P-L and P-W relations. Further investigation into the relative sensitivities of the three relations is necessary. Non-Linearity of the P-L Relation in the LMC Previous analysis of OGLE and MACHO observations of Cepheids suggests that the P-L relation is non-linear [4] However, this data contains many more short-period Cepheids than long-period Cepheids. This has led some to question the validity of the results. It is supposed that Cepheids in the inner field of NGC 4258 and in several galaxies recently observed with the Hubble Space Telescope [2] are of the same metallicity as those in the LMC. If this assumption is correct, then we can perform F tests on a combination of data from these galaxies. This is worthwhile, as these other data sets contain enough long-period Cepheids that when combined with the LMC data, there is a more balanced ratio of short- to long-period Cepheids. We first analyze the LMC and NGC 4258 data, then the LMC and HST data. LMC + NGC 4258 OGLE II data taken from [3] F statistic = 21.9395 Probability < 0.0001 Conclusion: Non-Linear F statistic = 7.0748 Probability = 0.1318 Conclusion: Linear F statistic = 1.8557 Probability = 0.4164 Conclusion: Linear Conclusion Although the P-L relation looks unquestionably non-linear, it seems inconsistent with linear P-W and P-C relations. LMC + HST All HST galaxy data taken from [2] F statistic = 1.6624 Probability = 0.4518 Conclusion: Linear F statistic = 3.2432 Probability = 0.2653 Conclusion: Linear F statistic = 26.9159 Probability = 0.0365 Conclusion: Non-Linear Conclusion A non-linear P-W relation is inconsistent with linear P-L and P-C relations. Impact on Hubble’s Constant Edwin Hubble noticed that a galaxy’s radial velocity is proportional to its distance from Earth: v = H 0 d The constant of proportionality, H 0, is called Hubble’s Constant. Increasing the accuracy of the distance to a galaxy increases the accuracy of our estimate of H 0. The uncertainty of the current estimate of H 0 is over 10%, but if NGC 4258 is used as a calibrating galaxy, the uncertainty may be reduced to as little as 5% [1]. References [1] Macri, L.M., et al, 2008 AJ 652:1133 [2] Davis, Matthew, unpublished MSc thesis, San Diego State University [3] Udalski, A., et al, 1999 Acta Astronomica Vol. 49 pp. 223-317 [4] Kanbur, S. M., et al, PASP Vol. 119, Iss. 855, pp. 512-552 Investigating Non-Linearity in NGC 4258 and the Large Magellanic Cloud P. A. Thompson 1, S. M. Kanbur 2, and C. C. Ngeow 3 1 University of Rochester, 2 SUNY Oswego, 3 National Central University, Taiwan Acknowledgement The authors thank NSF OISE award 0755646 and the 2006 Chretien award of the American Astronomical Society.