2017年IOAA实测第1题-测量大麦云的距离
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IOAA赛场上通常是使用英文回答问题，所以建议各位用英文回答本题
英文原题
(D1) Calibrating the distance ladder to the LMC [75 marks]
An accurate trigonometric parallax calibration for Galactic Cepheids has long been sought, but is very difficult to achieve in practice. All known classical (Galactic) Cepheids are more than 250 pc away, therefore for direct distance estimates to achieve an uncertainty of up to 10%, parallax uncertainties of up to ±0.2 milliarcsec are needed, requiring space-based observations. The Hipparcos satellite reported parallaxes for 200
of the nearest Cepheids, but even the best of these had high uncertainties. Recent progress has come with the use of the Fine Guidance Sensor on HST where parallaxes (in many cases) accurate to better than ±10% were obtained for 10 Cepheids, spanning a range of periods from 3.7 to 35.6 days. These nearby Cepheids cover distances from about 300 to 560 pc.
The measured periods, P, and average magnitudes in V, K and I bands are given in Table 1 as well as the AV and AK for extinction in V and K bands, respectively. The measured parallaxes with their uncertainties are also given in milliarcsec (mas). All measured apparent magnitudes have negligible uncertainty.
Table 1: Periods and average apparent magnitudes of 5 Galactic Cepheids with accurate parallax measurements.
| P
(day) |
<V>
(mag) |
<K>
(mag) |
AV
(mag) |
AK
(mag) |
<I>
(mag) |
parallax
(mas) |
error
(mas) | |
|---|---|---|---|---|---|---|---|---|
| RT Aur | 3.728 | 5.464 | 3.925 | 0.20 | 0.02 | 4.778 | 2.40 | 0.19 |
| FF Aql | 4.471 | 5.372 | 3.465 | 0.64 | 0.08 | 4.510 | 2.81 | 0.18 |
| X Sgr | 7.013 | 4.556 | 2.557 | 0.58 | 0.07 | 3.661 | 3.00 | 0.18 |
| ζ Gem | 10.151 | 3.911 | 2.097 | 0.06 | 0.01 | 3.085 | 2.78 | 0.18 |
| l Car | 35.551 | 3.732 | 1.071 | 0.52 | 0.06 | 2.557 | 2.01 | 0.20 |
(D1.1) The observed correlation between the period of a Cepheid and its brightness is usually described by the so-called “Period-Luminosity (PL) relation”, where L∝Pβ. In fact, such a relation is normally expressed in terms of the period and absolute magnitude, instead of luminosity. Hereafter, we shall refer to the Period-Absolute magnitude relation as the conventionally named “PL relation”.
Use the data given in Table 1 to plot a suitable linear graph in order to derive the Cepheid PL relation for the V-band and K-band. You should plot each graph separately on different pieces of graph paper. Determine the slope of the line that best describesthe linear relation of the data. (You may find the relation$$\Delta(log_{10}x)\approx\frac{\Delta x}{xlog_e 10}$$useful) [36.5 Marks]
Any apparent differences in PL relations of stars in the different bands can be explained if one also considers differences in colour. Therefore, the PL relation is in fact a PLC (Period-Luminosity-Colour) relation. This is from the reddening effect, due to extinction being a function of wavelength, which can also vary among different Cepheids due to their different metallicities, foreground Interstellar Medium and dust.
A new reddening-free magnitude (or bandpass) called “Wesenheit” has been proposed that does not require the explicit information of the extinction of individual stars but uses colour information from the star itself to get rid of the effect. For example, WVI use V and I band photometry and is defined as
$$W_{VI}=V-[\frac{A_V}{E（V-I）}]（V-I），$$
$$=V-R_v（V-I）$$
where Rv depends on the reddening law. In this case, we shall take the value of Rv to be 2.45.
(D1.2) From the data given in Table 1, plot and derive the reddening-free PL relation using Wesenheit WVI magnitudes. Estimate the linear slope of the relation as well as its uncertainty. [14.5 Marks]
(D1.3) Next, we would like to use the newly-derived PL relations from question (D1.1) & (D1.2) to estimate the distance to the Large Magellenic Cloud (LMC) using periods and magnitudes of classical Cepheids in the LMC. In Table 2, the periods, average extinction-corrected apparent magnitudes, <Vcorr>, and Wesenheit WVI magnitudes are given.
Estimate the distance modulus, μ , to each star and then use all the information to derive the distance to the LMC (in parsecs) and its standard deviation for each band.
Compare if the derived distances are statistically different for the 2 bands (YES/NO).
Are the standard deviations of the estimated distances for 2 bands different (YES/NO)?
Based on this dataset, which band (V or Wesenheit) is more accurate in estimating the distance to the LMC? [24 Marks]
Table 2: Period, average extinction-corrected apparent magnitude, <Vcorr>, and average Wesenheit magnitude measurements of Cepheids in the LMC
| P
(day) |
<Vcorr>
mag |
<WVI>
mag | |
|---|---|---|---|
| HV12199 | 2.63 | 16.08 | 14.56 |
| HV12203 | 2.95 | 15.93 | 14.40 |
| HV12816 | 9.10 | 14.30 | 12.80 |
| HV899 | 30.90 | 13.07 | 10.97 |
| HV2257 | 39.36 | 12.86 | 10.54 |
中文翻译
测量大麦云的距离
天文学家致力于通过三角视差法精确测量银河系内造父变星的距离，但很难取得成效。所有已知的银河系内造父变星的距离尺度都大于 250pc，由此导致测量结果存在 10%的不确定度，为此引入视差的误差范围：±0.2 毫角秒。依巴谷卫星测量了 200 颗紧邻造父变星的视差，但仍然具有很大的不确定度。近日，哈勃空间望远镜采用合理手段后获得了 10 颗误差范围在±10%以内的造父变星的视差。这些造父变星的周期范围为 3.7 至 35.6 天，距离范围为 300 至 560pc。
表 1 中给出了周期 P，V、K、I 三个波段的平均视星等<V>、<K>和<I>，V、K两个波段的消光 AV和 AK，视差 parallax（单位为毫角秒 mas），和视差的误差（单位为毫角秒 mas）。表中所有平均视星等的测量误差忽略不计。
表 1：5 颗银河系造父变星的周期、平均视星等和视差。
| P
(day) |
<V>
(mag) |
<K>
(mag) |
AV
(mag) |
AK
(mag) |
<I>
(mag) |
视差
(mas) |
视差的误差
(mas) | |
|---|---|---|---|---|---|---|---|---|
| RT Aur | 3.728 | 5.464 | 3.925 | 0.20 | 0.02 | 4.778 | 2.40 | 0.19 |
| FF Aql | 4.471 | 5.372 | 3.465 | 0.64 | 0.08 | 4.510 | 2.81 | 0.18 |
| X Sgr | 7.013 | 4.556 | 2.557 | 0.58 | 0.07 | 3.661 | 3.00 | 0.18 |
| ζ Gem | 10.151 | 3.911 | 2.097 | 0.06 | 0.01 | 3.085 | 2.78 | 0.18 |
| l Car | 35.551 | 3.732 | 1.071 | 0.52 | 0.06 | 2.557 | 2.01 | 0.20 |
(D1.1) 通过观测，天文学家发现造父变星的周期和光度存在周光关系，即 L∝Pβ。但实际使用中，上述关系常用绝对星等随周期的对数的变化来表示。本题中，请用修正后的周光关系解答。
根据表 1 中的数据，把解题和画图过程中会用到的各物理量的名称和计算结果写在答题纸上，并在坐标纸上通过描点和线性拟合，分别画出 V 波段和 K 波段的修正周光关系图。不同波段的关系图画在不同坐标纸上。根据线性拟合的结果，得到斜率的最佳拟合值和误差范围。（解题过程中你可能用到的近似为：$$\Delta(log_{10}x)\approx\frac{\Delta x}{xlog_e 10}$$） [36.5 分]
恒星的周光关系在不同波段是存在差异的，这可以用颜色的差异来解释。因此，修正后的周光关系又可以看作是（周期的对数-绝对星等-颜色）关系。颜色的变化的表现是红化效应，它会引起消光，而消光值是一个与波长有关的函数。消光值可能由于不同的造父变星的金属丰度不同或者前景星际介质或尘埃不同而改变。
为此，我们引入一个新的不依赖红化的平均视星等值，称为“Wesenheit”。这个星等值可以不考虑单个恒星的消光值，而是通过恒星的颜色信息有效地避开消光，从而得到恒星在对应波段的平均视星等。以 VI 波段为例，这个平均视星等值 在 VI 波段的定义式为：
$$W_{VI}=V-[\frac{A_V}{E（V-I）}]（V-I），$$
$$=V-R_v（V-I）$$
其中，Rv 是红化率。对于 VI 波段，Rv 取 2.45。
(D1.2) 根据表 1 中给出的数据，利用平均视星等 ，描点并画出新的不依赖红化的周光关系图。根据你画出的图得出线性拟合对应的斜率及误差范围。 [14.5 分]
(D1.3) 接下来，我们将用(D1.1)和(D1.2)两问中分别得出的周光关系，通过大麦云中的造父变星的周期和星等，计算大麦云的距离。表 2 中给出了周期、经过消光改正后的平均视星等<Vcorr>和利用新方法得到的平均视星等 <WVI>。
计算每个天体的距离模数，并根据所有信息计算大麦云中各天体对应不同距离模数的距离（以秒差距为单位）和各波段的标准差，用英语说明你使用的计算标准差的方法是什么。
请判断你得到的两种距离在统计学上是否相同(YES/NO)？
判断你得到的两种距离对应的标准差在统计学上是否相同(YES/NO)？
如果标准差不同，利用哪个星等值得到的距离更准确？V 还是 Wesenheit？ [24 分]
表 2：大麦云中造父变星的周期、经过消光修正的平均视星等<Vcorr>和利用新方法得到的平均视星等 <WVI>。
| P
(day) |
<Vcorr>
mag |
<WVI>
mag | |
|---|---|---|---|
| HV12199 | 2.63 | 16.08 | 14.56 |
| HV12203 | 2.95 | 15.93 | 14.40 |
| HV12816 | 9.10 | 14.30 | 12.80 |
| HV899 | 30.90 | 13.07 | 10.97 |
| HV2257 | 39.36 | 12.86 | 10.54 |
