2016年IOAA理论第6题-造父变星的脉动

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英文题目

(T6) Cepheid Pulsations

The star 𝛽-Doradus is a Cepheid variable star with a pulsation period of 9.84 days. We make a simplifying assumption that the star is brightest when it is most contracted (radius being 𝑅₁ ) and it is faintest when it is most expanded (radius being 𝑅₂ ). For simplicity, assume that the star maintains its spherical shape and behaves as a perfect black body at every instant during the entire cycle. The bolometric magnitude of the star varies from 3.46 to 4.08. From Doppler measurements, we know that during pulsation the stellar surface expands or contracts at an average radial speed of 12.8 km s ⁻¹ . Over the period of pulsation, the peak of thermal radiation (intrinsic) of the star varies from 531.0 nm to 649.1 nm.

(T6.1) Find the ratio of radii of the star in its most contracted and most expanded states (𝑅₁/𝑅₂ ).

(T6.2) Find the radii of the star (in metres) in its most contracted and most expanded states (𝑅₁ and 𝑅₂ ).

(T6.3) Calculate the flux of the star, 𝐹₂ , when it is in its most expanded state.

(T6.4) Find the distance to the star, 𝐷_star , in parsecs

中文翻译

(T6)造父变星脉动

剑鱼座$$\beta$$是一颗造父变星，其脉动周期为9.84天。我们做一个简化的假设，即恒星在最收缩时（半径为$$R_1$$）最亮，在最膨胀时（半径为$$R_2$$）最暗。为了简便起见，假设恒星保持球形，在整个周期中的每一瞬间都表现为一个完美黑体。这颗恒星的全波段星等从3.46到4.08不等。从多普勒测量，我们知道在脉动期间，恒星表面以$$12.8\rm{km\cdot s}^{-1}$$的平均径向速度膨胀或收缩。在脉动期间，恒星的热辐射（本征）峰值从$$531.0 \rm{nm}$$到$$649.1 \rm{nm}$$不等。

(T6.1) 求恒星在其最收缩和最膨胀状态下的半径比（$$R_1/R_2$$）。

(T6.2) 找出恒星在其最收缩和最膨胀状态（$$R_1$$和$$R_2$$）下的半径（单位为米）。

(T6.3) 计算恒星处于最大膨胀状态时的通量$$F_2$$。

(T6.4) 计算这颗恒星的距离（单位为秒差距）。