astro-init - 用户贡献 [zh-cn]

Topic:V7ftow3jgql3co7n

2019-09-16T15:16:50Z

Sqr.pi（讨论 | 贡献）已评论"关于2.2.的英文解答"的话题(看了下9月14日的版本，英文解答2.2确实有点错乱，像是录入时看错行漏了一段。)

2016年IOAA理论第11题-引力波

2019-09-16T13:52:48Z

Sqr.pi：添加翻译

{{需要解答}}
==英文题目==
'''(T11) Gravitational Waves'''

The first signal of gravitational waves was observed by two advanced LIGO detectors at Hanford and
Livingston, USA in September 2015. One of these measurements (strain vs time in seconds) is shown in
the accompanying figure. In this problem, we will interpret this signal in terms of a small test mass 𝑚
orbiting around a large mass 𝑀 (i.e., 𝑚 ≪ 𝑀), by considering several models for the nature of the central
mass.

[[文件:IOAA2016T11.jpg|无框|左]]

The test mass loses energy due to the emission of gravitational waves. As a result the orbit keeps on
shrinking, until the test mass reaches the surface of the object, or in the case of a black hole, the innermost
stable circular orbit – ISCO – which is given by 𝑅ISCO = 3𝑅sch, where 𝑅sch is the Schwarzschild
radius of the black hole. This is the “epoch of merger". At this point, the amplitude of the gravitational
wave is maximum, and so is its frequency, which is always twice the orbital frequency. In this problem,
we will only focus on the gravitational waves before the merger, when Kepler’s laws are assumed to be
valid. After the merger, the form of gravitational waves will drastically change.

(T11.1) Consider the observed gravitational waves shown in the figure above. Estimate the time period,
𝑇0
, and hence calculate the frequency, 𝑓0
, of gravitational waves just before the epoch of
merger.

(T11.2) For any main sequence (MS) star, the radius of the star, 𝑅MS, and its mass, 𝑀MS, are related by
a power law given as,

$$\begin{align*}
R_{MS}&\propto(M_{MS})^\alpha\\
where\ \alpha&=0.8\ for\ M_⊙\lt M_{MS}\\
&=1.0\ for\ 0.08M_⊙\leq M_{MS}\leq M_⊙
\end{align*} $$

If the central object were a main sequence star, write an expression for the maximum frequency
of gravitational waves, 𝑓MS, in terms of mass of the star in units of solar masses (𝑀MS/𝑀⊙)
and 𝛼.

(T11.3) Using the above result, determine the appropriate value of 𝛼 that will give the maximum
possible frequency of gravitational waves, 𝑓MS,max for any main sequence star. Evaluate this
frequency.

(T11.4) White dwarf (WD) stars have a maximum mass of 1.44 𝑀⊙ (known as the Chandrasekhar limit)
and obey the mass-radius relation 𝑅 ∝ 𝑀−1/3
. The radius of a solar mass white dwarf is equal to 6000 km. Find the highest frequency of emitted gravitational waves, 𝑓WD,max , if the test
mass is orbiting a white dwarf.

(T11.5) Neutron stars (NS) are a peculiar type of compact objects which have masses between
1 and 3𝑀⊙ and radii in the range 10 − 15 km. Find the range of frequencies of emitted
gravitational waves, 𝑓NS,min and 𝑓NS,max, if the test mass is orbiting a neutron star at a distance
close to the neutron star radius.

(T11.6) If the test mass is orbiting a black hole (BH), write the expression for the frequency of emitted
gravitational waves, 𝑓BH, in terms of mass of the black hole, 𝑀BH , and the solar mass 𝑀⊙.

(T11.7) Based only on the time period (or frequency) of gravitational waves before the epoch of merger,
determine whether the central object can be a main sequence star (MS), a white dwarf (WD), a
neutron star (NS), or a black hole (BH). Tick the correct option in the Summary Answersheet.
Estimate the mass of this object, 𝑀obj, in units of 𝑀⊙.

==中文翻译==

'''(T11) 引力波'''

2015年9月，两个先进的LIGO探测器在美国汉福德和利文斯顿观测到了首个引力波信号。其中一个测量值（应变与时间（单位为秒））如下图所示。在这个问题中，我们将通过考虑中心质量性质的几个模型，用一个质量为$$m$$的试验质量围绕质量为$$M$$的大质量物体（$$m \ll M$$）旋转来解释这一信号。

[[文件:IOAA2016T11.jpg|无框|左]]

由于引力波的发射，试验质量会损失能量。因此，其轨道继续收缩，直到测试质量到达大质量物体表面，或者在大质量物体是黑洞的情况下，达到最内部稳定的圆轨道（ISCO，由$$R_{\rm ISCO}=3R_{\rm sch}$$给出，其中$$R_{\rm sch}$$是黑洞的史瓦西半径）。这是“合并时期”。在这时，引力波的振幅是最大的，它的频率也是最大的，这一频率总是轨道频率的两倍。在这个问题中，我们只关注合并前的引力波，假设开普勒定律成立。合并后，引力波的形式将发生巨大变化。

(T11.1) 考虑上图所示的观测引力波。估计时间周期$$T_0$$，然后计算合并前引力波的频率$$f_0$$。

(T11.2) 对于任何主序星（MS），恒星的半径$$R_{\rm MS}$$及其质量$$M_{\rm MS}$$有幂律关系，如下所示，

\[
\begin{align*}
R_{\rm MS} & \propto (M_{\rm MS})^\alpha,\\
{其中，} \alpha & =
\begin{cases}
0.8 & M_\odot<M_{\rm MS},\\
1.0 & 0.08M_{\rm MS}\le M_{\rm MS} \le M_\odot.
\end{cases}
\end{align*}
\]

如果中心天体是主序星，用恒星质量（以太阳质量$$M_{\rm MS}/M_\odot$$为单位）和$$\alpha $$来表示引力波的最大频率$$f_{\rm MS}$$。

(T11.3) 使用上述结果，确定$$\alpha $$的值，使得主序星的引力波频率达到最大值$$f_{\rm MS,\ max}$$。计算这个频率。

(T11.4) 白矮星（WD）的最大质量为$$1.44 M_\odot$$（称为钱德拉塞卡极限），并服从质量-半径关系$$R\propto M^{-1/3}$$。太阳质量白矮星的半径等于$$6000 \rm{km}$$。如果试验质量围绕白矮星运行，求出发射引力波的最高频率$$f_{\rm WD,\ max}$$。

(T11.5) 中子星（NS）是一种特殊类型的致密天体，其质量在$$1\sim 3 M_\odot$$之间，半径在$$10\sim 15\rm{km}$$范围内。如果试验质量绕中子星运行的距离接近中子星半径，求出发射引力波的频率范围$$f_{\rm NS,\ max}$$和$$f_{\rm NS,\ max}$$。

(T11.6) 如果试验质量围绕黑洞（BH）运行，用黑洞的质量$$M_{\rm BH}$$和太阳质量$$M_\odot$$，表示所发射引力波的频率$$f_{\rm BH}$$。

(T11.7) 仅根据合并时期之前引力波的时间周期（或频率），确定中心天体是否可以是主序星（MS）、白矮星（WD）、中子星（NS）或黑洞（BH）。在答案汇总表中选择正确的选项。估计这个物体的质量$$M_{\rm obj}$$，单位是$$M_\odot$$。

[[分类:引力波]]

用户:Sqr.pi

2019-09-16T02:57:27Z

Sqr.pi：创建页面，内容为“根派，CNAO2013铁牌，$$\TeX$$爱好者，格式强迫症，现就读于复旦大学大数据学院。 [https://weibo.com/sqrpi 新浪微博]”

根派，CNAO2013铁牌，$$\TeX$$爱好者，格式强迫症，现就读于复旦大学大数据学院。

[https://weibo.com/sqrpi 新浪微博]

2016年IOAA理论第10题-引力透镜望远镜

2019-09-15T12:19:52Z

Sqr.pi：

{{需要解答}}
==英文题目==
(T10) Gravitational Lensing Telescope

Einstein's General Theory of Relativity predicts bending of light around massive bodies. For simplicity,
we assume that the bending of light happens at a single point for each light ray, as shown in the figure.
The angle of bending, 𝜃b , is given by

<nowiki>$$\theta_b =\frac{2R_{sch}}{r}$$</nowiki>

where 𝑅sch is the Schwarzschild radius associated with that gravitational body. We call 𝑟, the distance of
the incoming light ray from the parallel 𝑥-axis passing through the centre of the body, as the “impact
parameter”.

[[文件:IOAA2016T10.jpg|无框]]

A massive body thus behaves somewhat like a focusing lens. The light rays coming from infinite distance
beyond a massive body, and having the same impact parameter 𝑟, converge at a point along the axis, at a
distance 𝑓𝑟
from the centre of the massive body. An observer at that point will benefit from huge
amplification due to this gravitational focusing. The massive body in this case is being used as a
Gravitational Lensing Telescope for amplification of distant signals.

(T10.1) Consider the possibility of our Sun as a gravitational lensing telescope. Calculate the shortest
distance, 𝑓min, from the centre of the Sun (in A. U.) at which the light rays can get focused.

(T10.2) Consider a small circular detector of radius 𝑎, kept at a distance 𝑓min centered on the 𝑥-axis and
perpendicular to it. Note that only the light rays which pass within a certain annulus (ring) of
width ℎ (where ℎ ≪ 𝑅⊙) around the Sun would encounter the detector. The amplification factor
at the detector is defined as the ratio of the intensity of the light incident on the detector in the
presence of the Sun and the intensity in the absence of the Sun.

Express the amplification factor, 𝐴m, at the detector in terms of 𝑅⊙ and 𝑎.

(T10.3) Consider a spherical mass distribution, such as dark matter in a galaxy cluster, through which
light rays can pass while undergoing gravitational bending. Assume for simplicity that for the
gravitational bending with impact parameter, 𝑟, only the mass 𝑀(𝑟) enclosed inside the radius
𝑟 is relevant.

What should be the mass distribution, 𝑀(𝑟), such that the gravitational lens behaves like an ideal optical
convex lens?

==中文翻译==

'''(T10) 引力透镜望远镜'''

爱因斯坦的广义相对论预言了光在大质量物体周围的弯曲。为了简便起见，我们假设每个光线的弯曲都发生在一个点上，如图所示。弯曲角度$$\theta_b$$由下式给出

\[
\theta_b = \frac {2R_{\rm sch}} r
\]

其中$$R_{\rm sch}$$是与引力体相关的史瓦西半径。我们把入射光线与穿过物体中心的平行轴的距离$$r$$称为“碰撞参数”。

[[文件:IOAA2016T10.jpg|无框]]

因此，一个巨大的物体的行为有点像一个凸透镜。在大质量物体之外，来自无限远处，具有相同碰撞系数$$r$$的光线聚焦在光轴的同一点上，这一点距离大质量物体中心的距离为$$f_r$$。在这一点上的观测者将受益于引力聚焦带来的巨大放大效应。在这种情况下，这个巨大的天体被用作一个引力透镜望远镜，用来放大遥远的信号。

(T10.1) 考虑将我们的太阳作为引力透镜望远镜的可能性。计算光线可以被聚焦时距离太阳的最短距离$$f_{\rm min}$$。

(T10.2) 考虑一个半径为$$a$$的较小的圆形探测器，保持在最短距离$$f_{\rm min}$$处，以x轴为中心且垂直于x轴。注意，只有通过在围绕太阳的某个宽度为$$h$$（$$h \ll R_\odot$$）的环内的光线才会通过探测器。探测器处的放大系数被定义为在有太阳的情况下入射到探测器上的光的强度与在没有太阳的情况下的强度之比。

用$$R_\odot$$和$$a$$表示探测器的放大率$$A_{\rm m}$$。

(T10.3) 考虑一个球形的质量分布，例如星系团中的暗物质，光线在经历引力弯曲时可以通过它。为简便起见，假设对于具有碰撞参数$$r$$的引力弯曲，只考虑半径$$r$$内的质量$$M(r)$$。

如果引力透镜表现地和理想凸透镜一样，那么质量分布$$M(r)$$应该是怎样的？

[[分类:引力透镜]]

2016年IOAA理论第9题-火星环绕任务

2019-09-15T11:26:39Z

Sqr.pi：

{{需要解答}}
==英文题目==
(T9) Mars Orbiter Mission

India's Mars Orbiter Mission (MOM) was launched using the Polar Satellite Launch Vehicle (PSLV) on
5 November 2013. The dry mass of MOM (body + instruments) was 500 kg and it carried fuel of mass
852 kg. It was initially placed in an elliptical orbit around the Earth with perigee at a height of 264.1 km
and apogee at a height of 23903.6 km, above the surface of the Earth. After raising the orbit six times,
MOM was transferred to a trans-Mars injection orbit (Hohmann orbit).

The first such orbit-raising was performed by firing the engines for a very short time near the perigee.
The engines were fired to change the orbit without changing the plane of the orbit and without changing
its perigee. This gave a net impulse of 1.73 × 105kg m s
−1
to the satellite. Ignore the change in mass due
to burning of fuel.

(T9.1) What is the height of the new apogee, ℎa
above the surface of the Earth, after this engine
burn?

(T9.2) Find the eccentricity (𝑒) of the new orbit after the burn and the new orbital period (𝑃) of MOM
in hours.

==中文翻译==

'''(T9) 火星环绕任务'''

火星轨道器任务（MOM）于2013年11月5日使用极轨卫星运载火箭（PSLV）发射。MOM（箭体+仪器）的干质量为$$500\rm{kg}$$，携带的燃料质量为$$852\rm{kg}$$。它最初被发射在环绕地球的椭圆轨道上，近地点高度为距地球表面$$264.1\rm{km}$$，远地点高度为$$23903.6\rm{km}$$。在六次提升轨道后，MOM转移到一个火星转移轨道（霍曼轨道）。

第一次这样的轨道提升是通过在近地点附近很短时间内点燃发动机来完成的。点燃发动机是为了在不改变轨道平面和近地点的情况下改变轨道。这给卫星带来了$$1.73×10^5\rm{kg}\cdot \rm m \cdot s^{-1}$$的净冲量。忽略燃料燃烧引起的质量变化。

(T9.1) 发动机点燃后，新的远地点，距离地球表面的高度$$h_a$$是多少？

(T9.2) 求MOM新轨道的偏心率$$e$$和轨道周期$$P$$（单位为小时）。

[[分类:天体力学]]

2016年IOAA理论第9题-火星环绕任务

2019-09-15T11:25:53Z

Sqr.pi：

{{需要解答}}
==英文题目==
(T9) Mars Orbiter Mission

India's Mars Orbiter Mission (MOM) was launched using the Polar Satellite Launch Vehicle (PSLV) on
5 November 2013. The dry mass of MOM (body + instruments) was 500 kg and it carried fuel of mass
852 kg. It was initially placed in an elliptical orbit around the Earth with perigee at a height of 264.1 km
and apogee at a height of 23903.6 km, above the surface of the Earth. After raising the orbit six times,
MOM was transferred to a trans-Mars injection orbit (Hohmann orbit).

The first such orbit-raising was performed by firing the engines for a very short time near the perigee.
The engines were fired to change the orbit without changing the plane of the orbit and without changing
its perigee. This gave a net impulse of 1.73 × 105kg m s
−1
to the satellite. Ignore the change in mass due
to burning of fuel.

(T9.1) What is the height of the new apogee, ℎa
above the surface of the Earth, after this engine
burn?

(T9.2) Find the eccentricity (𝑒) of the new orbit after the burn and the new orbital period (𝑃) of MOM
in hours.

==中文翻译==

'''(T9) 火星环绕任务'''

火星轨道器任务（MOM）于2013年11月5日使用极轨卫星运载火箭（PSLV）发射。MOM（箭体+仪器）的干质量为$$500\rm{kg}$$，携带的燃料质量为$$852\rm{kg}$$。它最初被发射在环绕地球的椭圆轨道上，近地点高度为距地球表面$$264.1\rm{km}$$，远地点高度为$$23903.6\rm{km}$$。在六次提升轨道后，MOM转移到一个火星转移轨道（霍曼轨道）。

第一次这样的轨道提升是通过在近地点附近很短时间内点燃引擎来完成的。点燃发动机是为了在不改变轨道平面和近地点的情况下改变轨道。这给卫星带来了$$1.73×10^5\rm{kg}\cdot \rm m \cdot s^{-1}$$的净冲量。忽略燃料燃烧引起的质量变化。

(T9.1) 发动机点燃后，新的远地点，距离地球表面的高度$$h_a$$是多少？

(T9.2) 求MOM新轨道的偏心率$$e$$和轨道周期$$P$$（单位为小时）。

[[分类:天体力学]]

2016年IOAA理论第8题-U波段测光

2019-09-15T11:11:09Z

Sqr.pi：

{{需要解答}}
==英文题目==
'''(T8) U-Band photometry'''

A star has an apparent magnitude 𝑚U = 15.0 in the U-band. The U-band filter is ideal, i.e., it has perfect
(100%) transmission within the band and is completely opaque (0% transmission) outside the band. The
filter is centered at 360 nm, and has a width of 80 nm. It is assumed that the star also has a flat energy
spectrum with respect to frequency. The conversion between magnitude, 𝑚, in any band and flux density,
𝑓, of a star in Jansky (1 Jy = 1 × 10−26 W Hz
−1 m−2
) is given by

𝑓 = 3631 × 10−0.4𝑚Jy

(T8.1) Approximately how many U-band photons, 𝑁0
, from this star will be incident normally on a
1 m2
area at the top of the Earth's atmosphere every second?

This star is being observed in the U-band using a ground based telescope, whose primary mirror has a
diameter of 2.0 m. Atmospheric extinction in U-band during the observation is 50%. You may assume
that the seeing is diffraction limited. Average surface brightness of night sky in U-band was measured to
be 22.0 mag/arcsec2
.

(T8.2) What is the ratio, 𝑅, of number of photons received per second from the star to that received
from the sky, when measured over a circular aperture of diameter 2′′?

(T8.3) In practice, only 20% of U-band photons falling on the primary mirror are detected. How many
photons, 𝑁t
, from the star are detected per second?

==中文翻译==

'''(T8) U波段测光'''

恒星在U波段的视星等为$$m_{\rm U} = 15.0$$。U波段是理想的，也就是说，这一波段的透明度是完美的（100%）并且在该波段外完全不透光（透明度0%）。滤波器的中心是$$360 \rm{nm}$$，宽度为$$80 \rm{nm}$$。假设恒星在这一频率的能谱是平坦的。在任意波段下，星等$$m$$和恒星通量密度$$f$$（单位为央斯基，$$1 \rm{Jy} = 1\times 10^{-26} \rm{W} \cdot \rm{Hz}^{-1} \cdot \rm{m} ^{-2} $$）的转换关系为
\[
f = 3631 \times 10^{-0.4m} \rm{Jy}
\]

(T8.1) 每秒来自这颗恒星的U波段光子垂直入射到地球大气层顶部$$1 \rm m^2$$的区域的数目$$N_0$$大约为多少？

这颗恒星是用一台直径为$$2.0 \rm m$$的地面望远镜在U波段观测到的。观测期间U波段的大气消光率为50%。假设视宁度达到衍射极限。实测U波段夜空表面平均亮度为22.0等。

(T8.2) 当从一个直径为2"的圆形小孔中观测时，每秒从恒星接收到的光子数与从天空接收到的光子数之比$$R$$是多少？

(T8.3) 实际上，只有20%的U波段光子落在主镜上。每秒检测到的来自恒星的光子的数目$$N_{\rm t}$$是多少？

[[分类:星等]]

2016年IOAA理论第7题-望远镜光学结构

2019-09-15T04:52:36Z

Sqr.pi：add translation

{{需要解答}}
==英文题目==
'''(T7) Telescope optics'''

In a particular ideal refracting telescope of focal ratio 𝑓/5, the focal length of the objective lens is 100 cm
and that of the eyepiece is 1 cm.

(T7.1) What is the angular magnification, 𝑚0
, of the telescope? What is the length of the telescope,
𝐿0
, i.e. the distance between its objective and eyepiece?

An introduction of a concave lens (Barlow lens) between the objective lens and the prime focus is a
common way to increase the magnification without a large increase in the length of the telescope. A
Barlow lens of focal length 1 cm is now introduced between the objective and the eyepiece to double the
magnification.

(T7.2) At what distance, 𝑑B, from the prime focus must the Barlow lens be kept in order to obtain this
desired double magnification?

(T7.3) What is the increase, Δ𝐿, in the length of the telescope?

A telescope is now constructed with the same objective lens and a CCD detector placed at the prime focus
(without any Barlow lens or eyepiece). The size of each pixel of the CCD detector is 10 µm.

(T7.4) What will be the distance in pixels between the centroids of the images of the two stars , 𝑛p
, on
the CCD, if they are 20′′ apart on the sky?

==中文翻译==

'''(T7) 望远镜光学结构'''

在一个焦比为$$f/5$$的理想折射望远镜中，物镜焦距为$$100 \rm{cm}$$，目镜焦距为$$1 \rm{cm}$$。

(T7.1) 望远镜的角度放大倍数$$m_0$$是多少？望远镜的长度，即物镜和目镜之间的距离$$L_0$$是多少？

在物镜和原焦点之间引入凹透镜（巴洛镜）是在不大幅增加望远镜长度的情况下增加放大倍数的常用方法。现在在物镜和目镜之间引入焦距为$$1 \rm{cm}$$的巴洛镜，使放大倍率加倍。

(T7.2) 为了获得所需的双倍放大率，巴洛镜与原焦点的距离$$d_{\rm B}$$是多少？

(T7.3) 望远镜的长度增加值$$\Delta L$$是多少？

现在使用同样的物镜和一个CCD探测器来建造望远镜（不使用巴洛镜或目镜）。CCD探测器的每个像素的尺寸为$$10 \rm{\mu m}$$。

(T7.4) 如果两颗恒星在天空中相距20"，在CCD上的图像质心之间的像素距离$$n_p$$是多少？

[[分类:望远镜]]

2016年IOAA理论第6题-造父变星的脉动

2019-09-14T16:49:50Z

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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, 𝐷star , 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）。

[[分类:星等]]

2016年IOAA理论第5题-穿过GMRT波束

2019-09-12T17:11:38Z

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==英文题目==
'''(T5) GMRT beam transit'''

Giant Metrewave Radio Telescope (GMRT), one of the world's largest radio telescopes at metre
wavelengths, is located in western India (latitude: 19°6′ N, longitude: 74°3′ E). GMRT consists of 30
dish antennas, each with a diameter of 45.0 m. A single dish of GMRT was held fixed with its axis
pointing at a zenith angle of 39°42′ along the northern meridian such that a radio point source would pass
along a diameter of the beam, when it is transiting the meridian.

What is the duration 𝑇transit for which this source would be within the FWHM (full width at half
maximum) of the beam of a single GMRT dish observing at 200 MHz?

'''Hint:''' The FWHM size of the beam of a radio dish operating at a given frequency corresponds to the
angular resolution of the dish. Assume uniform illumination.

==中文翻译==
'''(T5) 穿越GMRT波束'''

巨米波射电望远镜（GMRT）位于印度西部（北纬19°6′，东经74°3′），是世界上最大的米波射电望远镜之一。GMRT由30个望远镜组成，每个望远镜的直径为$$45.0\rm m$$。GMRT的单个望远镜固定在其轴上，轴线指向天顶距39°42′的北子午线，这样当点状无线源通过子午线时，它将沿着波束的直径通过。

在$$200\rm {MHz}$$下，单个GMRT望远镜能观察到的半峰全宽范围内，该信号源的渡越持续时间$$T_{\rm transit}$$是多少？

'''提示：'''在给定频率下工作的无线电望远镜波束的半峰全宽与望远镜的角分辨率有关。假设均匀照明。

[[分类:视运动]]

2016年IOAA理论第4题-影子

2019-09-12T16:14:31Z

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==英文题目==

'''(T4) Shadows'''

An observer in the northern hemisphere noticed that the length of the shortest shadow of a 1.000 m
vertical stick on a day was 1.732 m. On the same day, the length of the longest shadow of the same
vertical stick was measured to be 5.671 m.

Find the latitude, 𝜙, of the observer and declination of the Sun, 𝛿⊙, on that day. Assume the Sun to be a
point source and ignore atmospheric refraction.

==中文翻译==

北半球的一位观测者注意到，一根$$1.000\rm m$$长的竖直棍子在一天中的最短阴影长度为$$1.732\rm m$$。同一天，同一根竖直棍子的最长阴影长度为$$5.671\rm m$$。

计算当天观测者的纬度$$\phi $$和太阳的赤纬$$\delta_{\odot}$$。假设太阳是点光源，忽略大气折射。

[[分类:视运动]]

2016年IOAA理论第4题-影子

2019-09-12T16:10:51Z

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2016年IOAA理论第3题-早期宇宙

2019-09-12T16:08:34Z

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==英文原题==
'''(T3) Early Universe'''

Cosmological models indicate that radiation energy density, 𝜌r
, in the Universe is proportional to
(1 + 𝑧)
4
, and the matter energy density, 𝜌m, is proportional to (1 + 𝑧)
3
, where 𝑧 is the redshift. The
dimensionless density parameter, Ω, is given as Ω = 𝜌/𝜌c
, where 𝜌c
is the critical energy density of the
Universe. In the present Universe, the density parameters corresponding to radiation and matter, are
Ωr0
= 10−4
and Ωm0
= 0.3, respectively.

(T3.1) Calculate the redshift, 𝑧e
, at which radiation and matter energy densities were equal.

(T3.2) Assuming that the radiation from the early Universe has a blackbody spectrum with a
temperature of 2.732 K, estimate the temperature, 𝑇e
, of the radiation at redshift 𝑧e
.

(T3.3) Estimate the typical photon energy, 𝐸ν
(in eV), of the radiation as emitted at redshift 𝑧e
.

==中文翻译==
'''(T3)早期宇宙'''

宇宙学模型表明，宇宙中的辐射能量密度$$\rho_r$$与$$(1+z)^4$$成正比，物质能量密度$$\rho_m$$与$$(1+z)^3$$成正比，其中$$z$$是红移。无量纲密度常数$$\Omega $$定义为$$\Omega = \rho /\rho_c$$，其中$$\rho_c$$是宇宙的临界能量密度。在当前宇宙中，与辐射和物质相对应的密度参数分别为$$\Omega_{r0}=10^{-4}$$和$$\Omega_{m0}=0.3$$。

(T3.1) 计算在辐射和物质能量密度相等时的红移$$z_e$$。

(T3.2) 假设来自早期宇宙的辐射是温度的黑体光谱温度为$$2.732\rm{K}$$，估计红移$$z_e$$处的辐射温度$$T_e$$。

(T3.3) 估计红移为$$z_e$$时的典型光子能量$$E_v$$（单位为$$\rm{eV}$$）。

[[分类:宇宙学]]

2016年IOAA理论第2题-土卫六上的气体

2019-09-12T15:56:13Z

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==英文题目==
'''(T2) Gases on Titan'''

Gas particles in a planetary atmosphere have a wide distribution of speeds. If the r.m.s. (root mean square)
thermal speed of particles of a particular gas exceeds 1/6 of the escape speed, then most of that gas will
escape from the planet. What is the minimum atomic weight (relative atomic mass), 𝐴min, of an ideal
monatomic gas so that it remains in the atmosphere of Titan?

Given, mass of Titan 𝑀T = 1.23 × 1023 kg, radius of Titan 𝑅T = 2575 km, surface temperature of
Titan 𝑇T = 93.7 K.

==中文翻译==
'''(T2)土卫六上的气体'''

行星大气中的气体粒子的速度分布很广。如果某一特定气体粒子的均方根热速度超过逃逸速度的1/6，那么大部分气体都将逃逸出这个星球。对于理想单原子气体来说，如果要留在土卫六的大气中，它的最小原子量（相对原子质量）$$A_{\textrm{min}}$$是多少？

已知，土卫六的质量$$M_{\rm{T}} = 1.23\times 10^{23} \rm{kg}$$，半径$${R_\rm{T}} = 2575 \rm{km}$$，表面温度$$T_\rm{T} = 93.7\rm{K}$$。

[[分类:热学]]

2016年IOAA理论第1题-判断正误

2019-09-12T09:49:39Z

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==英文题目==
'''(T1) True or False'''

Determine if each of the following statements is True or False. In the Summary Answersheet, tick the
correct answer (TRUE / FALSE) for each statement. No justifications are necessary for this question.

(T1.1) In a photograph of the clear sky on a Full Moon night with a sufficiently long exposure, the
colour of the sky would appear blue as in daytime.

(T1.2) An astronomer at Bhubaneswar marks the position of the Sun on the sky at 05: 00 UT every day
of the year. If the Earth's axis were perpendicular to its orbital plane, these positions would trace
an arc of a great circle.

(T1.3) If the orbital period of a certain minor body around the Sun in the ecliptic plane is less than the
orbital period of Uranus, then its orbit must necessarily be fully inside the orbit of Uranus.

(T1.4) The centre of mass of the solar system is inside the Sun at all times.

(T1.5) A photon is moving in free space. As the Universe expands, its momentum decreases.

==中文翻译==

''' (T1)判断正误 '''

判断下列陈述的正误。在答案汇总表中勾选每个陈述的正确答案。这个问题不需要解释理由。

(T1.1) 在曝光时间足够长的晴朗满月夜晚天空的照片中，天空的颜色会像白天一样是蓝色的。

(T1.2) 布巴内什瓦尔的一位天文学家在一年中的每天的世界时05:00标记太阳在天空中的位置。如果地轴垂直于地球的轨道面，这些位置将构成大圆上的弧。

(T1.3) 如果黄道面上某个小体绕太阳的轨道周期小于天王星的轨道周期，那么它的轨道必然完全在天王星轨道内。

(T1.4) 太阳系的质量中心永远在太阳内部。

(T1.5) 一个光子在自由空间中运动。由于宇宙膨胀，其动量会减小。