AKOW 010 - 天文問答．引力篇

[WenXP]

在 AKOW 009, 衰仔華用了不足一小時的時間就幾乎將所有題目破解, 似在嘲笑題目太易...

今次選幾題難一點的, 大家可能要上網查一查資料才解答得了. 由於很多專有名詞小弟都不知如何譯做中文(加上我懶), 所以索性將英文題原原本本的抄上來.

接招~


1. Which of the following methods has led to the most discoveries of massive planets orbiting near their parent stars?

a) detecting the shift of the star』s position against the sky due to the planet』s gravitational pull
b) detecting the infrared light emitted by the planet
c) detecting the light reflected by the planet
d) detecting the gravitational effect of an orbiting planet by looking for the Doppler shifts in the star』s spectrum


2. What happens to the orbit of Mercury as the Sun loses mass?
a) Mercury moves out because it has too much orbital energy.
b) It moves in because it has too much orbital energy.
c) Mercury stays put.
d) It moves out because it has too much angular momentum.


3. How do we estimate the height of the tidal bulge raised by the Moon on the Earth? On which latitude on Earth, we experience the minimum tidal height (mid latitude or pole/equator)? And why?


4. How close to the Earth does the tidal bulge on the Moon equals the size of the Moon? This orbital radius is called the Roche limit. Does this limit depend on the mass of the Moon?


5. A star rotates at 10 day period, an exo-planet orbits around it at 7 day period. Tidal dissipation
a) spins up the star, pushes away the planet
b) spins down the star, pushes away the planet
c) spins up the star, pulls in the planet
d) spins down the star, pulls in the planet[/url]

[david_kmng]

1=d
2 is a bit difficult. I think it is d.
3. Pole. I do not know what is tidal bulge. To estimate the height, I guess you have to calculate the gravitational force at 3 locations: the nearest point to Moon on earth, the centre of Earth and the fartherest point to Moon on earth. Then the differences are the height of the tide.
4. Use the concept above. I do not know if it is related to the mass of the moon.
5=c.

[Wah!]

英文問題好深呢!
你唔識得翻譯,我地又點知佢問乜呢?
啞左...
:-X

[WenXP]

"Tidal bulge" 解作 "潮汐力引起的隆起部份".
有關潮汐力, 可參巧 AKOW 002 .

啟明兄要改答案嗎? ^_^

[Wah!]

1. km個答案好似幾有道理,我跟!
2. 我選a
3. 第一部份唔係好知問乜,第二部份我都選極點,因為地球自轉使赤道較多水.
4.唔知
5.我都覺係c,因為潮水拉粒星轉,又因為AKOW9的第4題,所以行星會近左.

[david_kmng]

No need to change my answers.

For Q1, method a is the oldest method but cannot apply to distant stars (because the drift amount will be too small). Method b is not a very good method. Usually we "suspect" that stars with "accretion disk" may host planets. These accretion disk may even shows rings structure, a possible result of cleaning up action by planets. But we have never directly seen the planets in IR wavelength.
Some red or brown dwarfs, however, have a possible planets shown in IR directly. But this method only applies to red or brown dwarfs. They are dim and had to observe in a great distance, even though they are numerous.

Method c is somehow similar to method b, but the star light is usually million times brighter than the light reflected by the planets. This method only successfully applies to our solar system and has not been used since the discovery of Pluto since 1930.

However, an alterative method of detecting the brightness drop of a star by the occulation of the planets seems a more promising method and some results have been obtained (i.e this method relies on the detection of light NOT reflected by a planet).

Method d is difficult to use due to technical difficulties in measuring the tiny change is the star's spectrum. However, in the past 10 years this became possible. This method can be applied to star far far away as it is not sensitive to the distance of the star. Planets in stars which are outside our galaxy can be found, in theory. Secondly, this method has no constrain on the brightness of the stars.

Q2 is difficult to answer, as I do not know what is and how to calculate the orbital energy. But if we think about the momentum it seems that my answer applies.

[WenXP]

orbital energy of Mercury = KE + PE
But the fact that the Sun is losing mass implies that energy conservation does not hold. Therefore I won't consider energy conservation but angular momentum conservation here.

I think the answer for Q2 is d.
Sun's spinning angular momentum decreases, so Mercury's orbiting angular momentum must increase. (Mercury's spinning angular momentum seems won't change due to resonance.)

[WenXP]

第5題大家都答對, 不說了.

先不論第4題.
第3題, 俾個tips: 白道與天赤道有一個不小的夾角啊!

[david_kmng]

Yes. There is an inclination of the orbit of moon to the orbit of the earth. The amount is about 5 degree 9 minute.

It is not a big value. Can I neglect that?

Rochi limit, if I remember correctly, relies on the density of the object. For an object, such as a satellite of Jupiter. If the density of the satellite equals the density of Jupiter, then the Rochi distance is about 2.4 times of the radius of Jupiter.

The above is purely from my memory and I need to make a calculation to verify it.

Hmmm... All I can remember is the formula of gravity F=GMm/r^2

Let me do the calculation.

[WenXP]

For Q3, what is the height h of the tidal bulge raised by Moon on Earth?

M = Earth's mass
R = Earth's radius
m = mass of Moon
d = Earth-Moon distance

Now,
tidal acceleration = 2GmR/d^3 (please refer to AKOW002)
Earth's gravitation = GM/R^2

Rough estimation of h:

h = (tidal acceleration / Earth's gravitation) R = 2m/M (R/d)^3 R

Standard model is:

h = m/M (R/d)^3 R

[WenXP]

For Q4,

the above equation becomes
h = M/m (r/d)^3 r

Here r = Moon's radius

Moon will break apart whenever h is bigger than r:

h = M/m (r/d)^3 r r
d (M/m)^1/3 r

Here d is the Roche Limit.

[WenXP]

第1題大家都答對了.

第3題第二部分兩位都答赤道位置. 可以稍為解釋一下嗎?