2017年IOAA实测第2题-寻找暗物质
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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
目录
英文原题
(D2) The search for dark matter [75 marks]
A low surface brightness galaxy (LSB) is a diffuse galaxy with a surface brightness that, when viewed from the Earth, is at least one magnitude lower than the ambient night sky.
Some of its matter is in the form of “baryonic” matter such as neutral hydrogen gas and stars. However, most of its matter is in the form of invisible mass – so called “dark matter”. In this question, we will investigate the mass of dark matter in a galaxy, the effect of dark matter on the rotation curves of the galaxy, and the distribution of dark matter in the galaxy.
The table below provides the data of a LSB galaxy named UGC4325. The galaxy is assumed to be edge-on. At every distance r from the centre of the galaxy, we measure
1.$$λ_{obs}$$ , the observed wavelength of the Hα emission line. The Hubble expansion of the Universe has already been excluded from the data.
2.$$V_{gas}$$ , the contribution of the gas component to the rotation due to $$M_{gas}$$ , derived from HI surface densities.
3.$$V_*$$ , the contribution of the stellar component to the rotation due to $$M_*$$ , derived from Rband photometry.
The rotational velocities of the test particle due to the gas component, $$V_{gas}$$ , and the star component, $$V_*$$ , are defined as the velocities in the plane of the galaxy that would result from the corresponding components without any external influences. These velocities are calculated from the observed baryonic mass density distributions.
| r
(kpc) |
$$λ_{obs}$$
(nm) |
$$V_{gas}$$
(km/s) |
$$V_*$$
(km/s) |
|---|---|---|---|
| 0.70 | 656.371 | 2.87 | 20.97 |
| 1.40 | 656.431 | 6.75 | 32.22 |
| 2.09 | 656.464 | 14.14 | 40.91 |
| 2.79 | 656.475 | 20.18 | 46.75 |
| 3.49 | 656.478 | 24.08 | 50.10 |
| 4.89 | 656.484 | 28.08 | 47.94 |
| 6.25 | 656.481 | 29.25 | 45.47 |
| 7.10 | 656.481 | 27.03 | 47.78 |
| 9.03 | 656.482 | 25.90 | 45.32 |
| 12.05 | 656.482 | 21.03 | 42.30 |
The mass of dark matter $$M_{DM} (r)$$ within a volume of radius $$r$$ can be defined in terms of the rotational velocity due to dark matter $$V_{DM}$$, the radius $$r$$ and gravitational constant $$G$$ ,
To a good approximation, the observed rotational velocity $$V_{obs}$$ can be modelled as
The observed rotational velocity $$V_{obs}$$ depends on the mass of the galaxy $$M(r)$$ within a volume of radius $$r$$ measured from the galaxy’s centre.
The mass density $$ρ_{DM}(r)$$ of dark matter within a volume of radius $$r$$ can be modelled by a galaxy density profile,
where $$ρ_0$$ and $$r_C$$ are the central density and the core radius of the galaxy, respectively. According to the density profile, the mass of dark matter $$M_{DM}(r)$$ within a volume of a radius $$r$$ can be described by
Part 1 The mass of dark matter and rotation curves of the galaxy
(D2.1) In laboratories on Earth, Hαhas an emitted wavelength $$λ_{emit}$$ of 656.281 nm. Compute the observed rotational velocities of the galaxy $$V_obs$$ and the rotational velocities due to the dark matter $$V_DM$$ at distance $$r$$ in units of km s-1.
For the different values of $$r$$ given in the table, compute the dynamical mass $$M(r)$$ and the mass of dark matter $$M_{DM}(r)$$ in solar masses. [21]
(D2.2) Create rotation curves of the galaxy on graph paper by plotting the points of $$V_{obs}$$ , $$V_{DM}$$ , $$V_{gas}$$ , $$V_*$$ versus the radius $$r$$ and draw smooth curves through the points (mark your graph as “D2.2”).
Order the contribution of the different components to the observed velocity in descending order. [16]
Part 2 Dark matter distribution
(D2.3) Take a data point at small $$r$$ and large $$r$$ to estimate $$ρ_0$$ and $$r_C$$ . Note that for large values of $$x$$ , $$arctan(x)\approxπ/2$$ and at small $$x$$ ,$$arctan(x)\approx x-{x^3}/3$$ . [7]
(D2.4) By comparing Equation (4) to a linear function, the central density $$ρ_0$$ could also be found by a linear fit. Plot an appropriate graph so that a linear fit can be used to find another value of $$ρ_0$$ . Evaluate $$ρ_0$$ in units of M⨀ kpc-3. (Mark your graph as “D2.4”). If you cannot find the value of $$r_C$$ from the previous part, use $$r_C$$ =3.2 kpc as an estimate for this part. [19]
(D2.5) Compute logarithmic values of the dark matter density, ln[ρDM(r)] , and plot the distribution of dark matter in the galaxy as a function of radius $$r$$ on graph paper.(Mark your graph as “D2.5”). [12]
中文翻译
寻找暗物质
低表面亮度星系（LSB）是一种弥散星系。当从地球观测时，这种星系的表面亮度至少比周围的夜天光低一个星等。星系中的一部分物质是以中性氢气体和恒星为代表的“重子物质”，而大部分是不可见的暗物质。本题我们将研究某个星系中的暗物质的质量对星系的旋转曲线的影响， 并描述该星系中的暗物质的相关情况。
下表是低表面亮度星系UGC4325的一些数据。如果假定这个星系是完全侧向我们的。相对于到星系中心的距离 $$r$$ ，我们定义如下参数。
1.$$λ_{obs}$$ , Hα 线的观测波长，宇宙哈勃膨胀的影响已被排除在外。
2.$$V_{gas}$$ ,由 HI 表面密度导致的气体质量 $$M_{gas}$$ 产生的速度对星系总的旋转速度的贡献。
3.$$V_*$$ , R波段测光得到的恒星质量 $$M_*$$ 产生的速度对星系总的旋转速度的贡献。
来自气体 $$V_{gas}$$ 和来自恒星 $$V_*$$ ,产生的速度对星系总的旋转速度的贡献被定义为星系盘在不考虑任何外部因素影响下，由于对应机制作用产生的速度。这些速度是通过观测重子物质的密度分布计算得出的。
| r
(kpc) |
$$λ_{obs}$$
(nm) |
$$V_{gas}$$
(km/s) |
$$V_*$$
(km/s) |
|---|---|---|---|
| 0.70 | 656.371 | 2.87 | 20.97 |
| 1.40 | 656.431 | 6.75 | 32.22 |
| 2.09 | 656.464 | 14.14 | 40.91 |
| 2.79 | 656.475 | 20.18 | 46.75 |
| 3.49 | 656.478 | 24.08 | 50.10 |
| 4.89 | 656.484 | 28.08 | 47.94 |
| 6.25 | 656.481 | 29.25 | 45.47 |
| 7.10 | 656.481 | 27.03 | 47.78 |
| 9.03 | 656.482 | 25.90 | 45.32 |
| 12.05 | 656.482 | 21.03 | 42.30 |
半径为 $$r$$ 范围内的星系所含的暗物质质量 $$M_{DM} (r)$$ 可以用暗物质贡献的旋转速度 $$V_{DM}$$, 半径 $$r$$ 和引力常数 $$G$$ 表示为,
现有的观测旋转速度 $$V_{obs}$$ 的最佳约束模型为：
对于到星系中心为 $$r$$ 的位置，该位置的旋转速度 $$V_{obs}$$ 取决于这个范围内的星系质量 $$M(r)$$。
根据星系密度分布模型，暗物质质量密度 $$ρ_{DM}(r)$$ 与到星系中心距离 $$r$$ 的关系为：
其中 $$ρ_0$$ 和 $$r_C$$ 分别为该星系的中心密度和核半径。
根据密度分布，暗物质的质量 $$M_{DM}(r)$$ 随到星系中心距离 $$r$$ 变化的关系由下式表述：
暗物质质量和星系的旋转曲线
(a) 地球上Hα的实验室发射波长 $$λ_{emit}$$ 为 656.281 nm. 计算在距离 $$r$$ 上观测到的星系旋转速度 $$V_obs$$ 以及暗物质贡献的旋转速度 $$V_DM$$ ,单位 km s-1.
通过表中给出的不同距离 $$r$$ 计算动力学总质量 $$M(r)$$ 和暗物质质量 $$M_{DM}(r)$$ ，单位为太阳质量。
(b) 在坐标纸上画出星系的旋转曲线。在同一坐标系中分别画出 $$V_{obs}$$ 、 $$V_{DM}$$ 、 $$V_{gas}$$ 、 $$V_*$$ 随 $$r$$ 的变化，标出计算出的速度点并用平滑曲线链接。 (标注你的图为 “D2.2”)
依据观测到的旋转速度，对不同物质的贡献进行排序。
暗物质的贡献
(c) 使用较大和较小的 $$r$$ 估算 $$ρ_0$$ 和 $$r_C$$ 。注意当 $$x$$ 很大时, $$arctan(x)\approxπ/2$$ ； 当$$x$$ 很小时,$$arctan(x)\approx x-{x^3}/3$$ .
(d) 把公式(4) $$M_{DM}(r)=4πρ_0r_C^2[r-r_Carctan(r/r_C)]$$ 看做一个线性方程,中心密度 $$ρ_0$$ 可以通过线性拟合得出。画合适的坐标图用于拟合另一个 $$ρ_0$$ 的值， 单位为 M⨀ kpc-3。(标注你的图为 “D2.4”). 如果你在之前的部分没有得到 $$r_C$$ 的值,那么在这里用 $$r_C$$ =3.2 kpc 进行估算。
(e)计算暗物质密度的对数值 $$ln[ρ_{DM}(r)] $$, 在坐标纸上画图呈现改星系中暗物质随到星系中心距离 $$r$$ 的变化趋势。(标注你的图为 “D2.5”).
（注释：
这里的翻译是结合了《天文爱好者》和原题给出的。但是总归还是要看原题比较保险。如果翻译里头没有翻译出“标注你的图为xxx”然后你愉快【一点也不】地漏了标注图，你的分数犹如风中残烛。）
