平行双步相位移显微干涉测量新方法：基于立方分束器的应用

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本文介绍了一种基于立方分束器的并行两步相位移显微干涉测量方法，即Parallel Two-step Phase-shifting Microscopic Interferometry (PPSI)。传统的相位移干涉技术通过在不同时间间隔对同一参考波进行相位调制，以便于复原物体的干涉图。然而，该新方法利用了45度倾斜的立方分束器的独特特性，使得物体和参考光波被复制成两个相互平行的光束。这使得每个光束上可以同时获得具有正交相位偏移的干涉图，从而能够并行地进行相位测量。通常，两步相位移干涉涉及到两次曝光，每次曝光时物体光经过不同的相位调制，例如经典的Shack-Hartmann法或Zernike波前传感器。而在这种新型PPSI中，由于两个光束的独立操作，可以同时获取两个相位差的干涉图像，这显著提高了数据采集效率，并可能减少系统复杂性和噪声影响。文章讨论了如何通过改进的算法对这两个并行干涉图进行相位重建，这可能包括利用数字信号处理技术，如自适应滤波或者利用快速傅里叶变换（FFT）来提取相位信息。这种方法对于高精度的应用，如显微镜成像、生物医学检测、纳米尺度测量以及光学计量学等领域有着潜在的优势，因为它允许实时的多通道数据采集和处理，从而节省了时间和资源。同时，文中还提到了与传统技术（如Simultaneous Phase-shifting Interferometry, SPSI 和 Digital Holography Microscopy, DHM）的比较，这些技术虽然在某些方面有其优势，但并行两步相位移方法在并行性和实时性方面表现出更强的竞争力。至于Cepheid变星研究中的应用，尽管这部分内容与论文标题的主题有所偏离，但值得注意的是，文章提及的测量恒星直径、校准勒维特定律（Leavitt Law）以及利用Baade-Wesselink方法确定距离的问题，这些都是在天体物理学中使用类似干涉技术的重要应用，特别是对于精确测量距离和理解宇宙膨胀的挑战。总结来说，本文不仅介绍了并行两步相位移显微干涉技术的新实现，而且还探讨了它在提高测量精度、速度和实用性方面的潜力，这在科学实验尤其是精密测量领域具有重要意义。同时，它也展示了这种技术在其他领域的应用前景，如天文学中的恒星参数测量。

arXiv:1203.3552v1 [astro-ph.SR] 15 Mar 2012

Astronomy & Astrophysics

manuscript no. proj2

 ESO 2012

March 19, 2012

Cepheid limb darkening, angular diameter corrections, and

projection factor from static spherical model stellar atmospheres

Hilding R. Neilson

, Nicolas Nardetto

, Chow-Choong Ngeow

, Pascal Fouqu´e

, and Jesper Storm

Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, D-53121 Bonn, Germany

e-mail: hneilson@astro.uni-bonn.de

Laboratoire Lagrange, UMR7293, UNS/CNRS/OCA , 06300 Nice, France

Graduate Institute of Astronomy, National Central University, Jhongli City, 32001, Taiwan

IRAP, Universit´e de Toulouse, CNRS, 14 avenue Edouard Belin, 31400 Toulouse, France

Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany

ABSTRACT

Context.

One challenge for measuring the Hubble constant using Classical Cepheids is the calibration of the Leavitt Law or period-

luminosity relationship. The Baade-Wesselink method for distance determination to Cepheids relies on the ratio of the measured radial

velocity and pulsation velocity, the so-called projection factor and the ability to measure the stellar angular diameters.

Aims.

We use spherically-symmetric model stellar atmospheres to explore the dependence of the p-factor and angular diameter cor-

rections as a function of pulsation period.

Methods.

Intensity proﬁles are computed from a grid of plane-parallel and spherically-symmetric model stellar atmospheres using

the SAtlas code. Projection factors and angular diameter corrections are determined from these intensity proﬁles and compared to

previous results.

Results.

Our predicted geometric period-projection factor relation including previously published state-of-the-art hydrodynamical

predictions is not with recent observational constraints. We suggest a number of potential resolutions to this discrepancy. The model

atmosphere geometry also aﬀects predictions for angular diameter corrections used to interpret interferometric observations, suggest-

ing corrections used in the past underestimated Cepheid angular diameters by 3 − 5%.

Conclusions.

While spherically-symmetric hydrostatic model atmospheres cannot resolve diﬀerences between projection factors from

theory and observations, they do help constrain underlying physics that must be included, including chromospheres and mass loss.

The models also predict more physically-based limb-darkening corrections for interferometric observations.

Key words. stars:atmospheres / stars:distances / stars: variable: Cepheids

1. Introduction

Classical Cepheids form an important rung on the cosmic

distance ladder because they follow a period-luminosity re-

lation, also called the Leavitt Law (Leavitt 1908). As such,

Cepheids are used to measure distances to other galax-

ies (e.g. Pietrzy´nski et al. 2010), and cosmological parame-

ters (Freedman et al. 2001; Riess et al. 2009, 2011), but the

Leavitt Law must be calibrated using Galactic (e.g. Turner

2010) and Large Magellanic Cloud (LMC) Cepheids (e.g.

Ngeow & Kanbur 2008; Ngeow et al. 2009).

The Leavitt Law is calibrated by measuring Cepheid dis-

tances using numerous of techniques, such as parallax, clus-

ter membership, and the Baade-Wesselink method. The Leavitt

Law has been derived by assuming all LMC Cepheids are

at the same distance, where the distance to the LMC is de-

rived using a number of methods (e.g. Gieren et al. 2005;

Clement et al. 2008; Bonanos et al. 2011). Parallax has been

measured for a number of Galactic Cepheids (van Leeuwen et al.

2007; Benedict et al. 2007), but this method is limited by the

precision of Hipparcos and the Hubble Space Telescope. Only

a small number of Cepheids have precise parallaxes measured,

and this will only change when the GAIA satellite is launched

(Windmark et al. 2011). Likewise, Turner (2010) measured dis-

tances to 24 Cepheids belonging to clusters and groups. On

the other hand, the Baade-Wesselink method can be used for

Galactic and LMC Cepheids (Kervella et al. 2004; Gieren et al.

2005), allowing for a much greater sample size.

The Baade-Wesselink method (Baade 1926; Wesselink

1946) measures the distance to a Cepheid by measuring the

change of angular diameter and the actual change of radius as

a function of phase. These measurements yield the mean radius

and mean angular diameter, hence the distance. The change of

angular diameter can be determined using a number of meth-

ods. One method is from interferometric observations, where the

angular diameter can be directly resolved (Kervella et al. 2004,

2006; M´erand et al. 2006). The method uses angular diameter

corrections for limb darkening when the observations have lim-

ited resolution. A second method is using the infrared surface

brightness technique (Barnes & Evans 1976; Gieren et al. 1989).

This method uses a correlation between the angular diameter at

any phase and the star’s V-band ﬂux and color. This method is

derived from the relation F

Bol

∝ θ

eﬀ

, where θ is the angu-

lar diameter, F

Bol

is the bolometric ﬂux and T

eﬀ

is the eﬀec-

tive temperature. The eﬀective temperature is correlated to the

color based on observations of other stars or model stellar atmo-

spheres, while the bolometric ﬂux is replaced by the V-band ﬂux

and the bolometric correction. This method assumes the radius

is deﬁned at a speciﬁc location in the stellar photosphere, i.e. at

an optical depth τ = 2/3.

The change of radius is ∆R =

puls

dt, where v

puls

is the

pulsation velocity. One cannot directly measure the pulsation ve-

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