Introduction

1. Investigating Non-Linearity in NGC 4258 and the Large Magellanic Cloud P. A. Thompson1, S. M. Kanbur2, and C. C. Ngeow3 1University of Rochester, 2SUNY Oswego, 3National Central University, Taiwan • 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 = H0d • The constant of proportionality, H0, is called Hubble’s Constant. • Increasing the accuracy of the distance to a galaxy increases the accuracy of our estimate of H0. • The uncertainty of the current estimate of H0 is over 10%, but if NGC 4258 is used as a calibrating galaxy, the uncertainty may be reduced to as little as 5% [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 – 5log10(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 = { a1 + b1logP, P < 10; a2 + b2logP, 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 = (RSSR – RSSF) × (vR– vF ) RSSR = residual sum of squares of linear model RSSF vF RSSF = RSS of non-linear model vR = degrees of freedom of linear model (N – 2) vF = 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. • 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 Acknowledgement The authors thank NSF OISE award 0755646 and the 2006 Chretien award of the American Astronomical Society.