2017年IOAA理论第2题-地球凌日带

来自astro-init

Search-line.png

  • 需要解答

本题目目前没有解答。要不要你来试试!


英文原题

(T2) Earth's Transit Zone [10 marks]

Earth's transit zone is an area where extrasolar observers (located far away from the Solar System) can detect the Earth transiting across the Sun. For observers on the Earth, this area is the projection of a band around the Earth's ecliptic onto the celestial plane (light grey area in the left figure). Assume that the Earth has a circular orbit of 1 au.

2017ioaaT2 Q1.png

a) Find the angular width of that part of the Earth's transit zone in degrees, in which the extrasolar observers can detect Earth's total transit (when the whole of the Earth's disk passes in front of the Sun).

[5]


b) Find the angular width of that part of the Earth's transit zone in degrees, where the extrasolar observers can detect at least Earth's grazing transit (when any part of the Earth's disk passes in front of the Sun).

[5]

中文翻译

地球凌日带

“地球凌日带”是太阳系外的观测者(在离太阳系很远的地方)能够观测到地球凌日的区域。在地球上看,这个区域是地球的影子在黄道附近形成的条带(即左图浅灰色区域)。假定地球的轨道是正圆,半径为1 au。

(a)计算太阳系外的观测者可以观测到地球全凌(也就是整个地球圆面从太阳前穿过)的地球凌日带宽度,以角度为单位。

(b)计算太阳系外的观测者最少可以观测到地球掠凌(部分地球圆面从太阳前穿过)的地球凌日带宽度,以角度为单位。

官方解答

英文原文

(a) For Earth's transit, the whole Earth's disk should pass in front of the sun,

IOAA2017 T2-1.png

From ΔSTA~ΔETD:
[1 mark]


$$\frac{X}{R_⊕}=\frac{a+X}{R_T}$$


$$X=\frac{aR_⊕}{R_T-R_⊕}$$
[1 mark]


The half angular size of the Earth’s Transit Zone with transit can be written as,


$$\theta_T=arcsin(\frac{R_⊕}{X})+arcsin(\frac{R_T}{a+X})=arcsin(\frac{R_T-R_⊕}{a})$$
[1 mark]


Solution with$$\theta_T=arcsin(\frac{R_T-R_⊕}{a})\approx\frac{R_T-R_⊕}{a} $$is acceptable with full mark

The angular size of the Earth’s Transit Zone with transit is

$$2\theta_T=2arcsin(\frac{R_T-R_⊕}{a})$$
[1mark]


$$2\theta_T=0.527°$$
[1mark]