2016年IOAA理论第12题-Astrosat(Discarded)

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英文题目

(T12) AstroSat

India astronomy satellite, AstroSat, launched in September 2015, has five different instruments.

In this question, we will discuss three of these instruments (SXT, LAXPC, CZTI), which point in the same direction and observe in X-ray wavelengths. The details of these instruments are given in the table below.

Instrument	Band [keV]	Collecting Area [m2 ]	Effective Photon Detection Efficiency	Saturation level [counts]	No. of Pixels
SXT	0.3-80	0.067	60%	1500(total)	512x512
LAXPC	3-80	1.5	40%	50000 (in any one counter) or 200000 (total)	---
CZTI	10-150	0.09	50%	---	4 x 4096

You should note that LAXPC energy range is divided into 8 different energy band counters of equal bandwidth with no overlap.

(T12.1) Some X-ray sources like Cas A have a prominent emission line at 0.01825 nm corresponding to a radioactive transition of ⁴⁴Ti . Suppose there exists a source which emits only one bright emission line corresponding to this transition. What should be the minimum relative velocity (𝑣) of the source, which will make the observed peak of this line to get registered in a different energy band counter of LAXPC as compared to a source at rest?

These instruments were used to observe an X-ray source (assumed to be a point source), whose energy spectrum followed the power law,

𝐹(𝐸) = 𝐾𝐸^−2⁄3 [in units of counts/keV/m²/s ]

where 𝐸 is the energy in keV, 𝐾 is a constant and 𝐹(𝐸) is photon flux density at that energy. Photon flux density, by definition, is given for per unit collecting area (m² ) per unit bandwidth (keV) and per unit time (seconds). From prior observations, we know that the source has a flux density of 10 counts/keV/m² /s at 1 keV, when measured by a detector with 100% photon detection efficiency. The “counts” here mean the number of photons reported by the detector. As the source flux follows the power law given above, we know that for a given energy range from 𝐸₁ (lower energy) to 𝐸₂ (higher energy) the total photon flux (𝐹_T) will be given by

𝐹_T = 3𝐾 (𝐸₂^1⁄3 − 𝐸₁^1⁄3) [in units of counts/m²/s ]

(T12.2) Estimate the incident flux density from the source at 1 keV, 5 keV, 40 keV and 100 keV. Also estimate what will be the total count per unit bandwidth recorded by each of the instruments at these energies for an exposure time of 200 seconds.

(T12.3) For this source, calculate the maximum exposure time (𝑡_S ), without suffering from saturation, for the CCD of SXT.

(T12.4) If the source became 3500 times brighter, calculate the expected counts per second in LAXPC counter 1, counter 8 as well as total counts across the entire energy range. If we observe for longer period, will the counter saturate due to any individual counter or due to the total count? Tick the appropriate box in the Summary Answersheet.

(T12.5) Assume that the counts reported by CZTI due to random fluctuations in electronics are about 0.00014 counts per pixel per keV per second at all energy levels. Any source is considered as “detected” when the SNR (signal to noise ratio) is at least 3. What is minimum exposure time, 𝑡_𝑐 , needed for the source above to be detected in CZTI? Note that the “noise” in a detector is equal to the square root of the counts due to random fluctuations.

(T12.6) Let us consider the situation where the source shows variability in number flux, so that the factor 𝐾 increases by 20%. AstroSat observed this source for 1 second before the change and 1 second after this change in brightness. Calculate the counts measured by SXT, LAXPC and CZTI in both the observations. Which instrument is best suited to detect this change? Tick the appropriate box in the Summary Answersheet.

中文翻译

印度的天文卫星AstroSat于2015年9月发射，配备了五种不同的仪器。在这个问题中，我们将讨论这三种仪器（SXT、LAXPC、CZTI），它们指向同一方向，并在X射线波段进行观测。下面的表格中给出了这些仪器的详细信息。

你应该注意，LAXPC 的能量范围被划分为 8 个不同的能量带计数器，带宽相等且没有重叠。

(T12.1)一些像 Cas A 的 X 射线源在 0.01825 nm处有一个显著的发射线，对应于 ⁴⁴Ti 的放射性跃迁。假设存在一个只发射这一条亮发射线的源。那么这个源的最小相对速度（𝑣）应该是多少，才能让这条线的观测峰值在 LAXPC 的不同能量段计数器中被记录，而不是像静止源那样？

这些仪器用来观测一个 X 射线源（假设为点源），它的能量谱遵循幂律，

𝐹(𝐸) = 𝐾𝐸^−2⁄3 [in units of counts/keV/m²/s ]

其中 𝐸 表示能量（单位为 keV），𝐾 是一个常数，𝐹(𝐸) 是该能量下的光子通量密度。光子通量密度按定义是每单位收集面积（m²）、每单位带宽（keV）和每单位时间（秒）给出的。从以前的观测中，我们知道当用一个光子探测效率为 100% 的探测器测量时，该源在 1 keV 处的通量密度是 10 counts/keV/m²/s。这里的“counts”指的是探测器报告的光子数量。由于源通量遵循上述幂律，我们就知道在一个能量范围从 𝐸1（低能）到 𝐸2（高能）时，总光子通量（𝐹T）将由下式给出

𝐹_T = 3𝐾 (𝐸₂^1⁄3 − 𝐸₁^1⁄3) [in units of counts/m²/s ]

(T12.2)估算在 1 keV、5 keV、40 keV 和 100 keV 时来自源的入射通量密度。同时估算在这些能量下，每个仪器在 200 秒的曝光时间内每单位带宽记录的总计数。

(T12.3) 对于这个光源，计算 SXT CCD 的最大曝光时间 (𝑡S)，以避免过曝。

(T12.4)如果光源变得亮了3500倍，计算LAXPC计数器1、计数器8以及整个能量范围总计数的每秒预期计数。如果观察时间更长，计数器会因为任何单个计数器或总计数而饱和吗？在总结答案表上勾选相应的选项。

(T12.5)假设CZTI报告的由于电子随机波动引起的计数在所有能量水平上约为每像素每keV每秒0.00014计数。当信号噪声比(SNR)至少为3时，任何源都被认为是“检测到”的。上面这个源在CZTI中被检测到所需的最短曝光时间𝑡𝑐是多少？

注意，探测器中的“噪声”等于由随机波动引起的计数的平方根。

(T12.6) 让我们考虑这样一种情况：源的计数通量显示出变化，使得因子𝐾增加了20%。AstroSat在亮度变化前观察了该源1秒，变化发生后又观察了1秒。计算SXT、LAXPC和CZTI在两次观测中测得的计数。哪台仪器最适合检测这一变化？在总结答案表中勾选相应的选项。