Short Problem Final

SHORT QUESTIONS 1. There are several exoplanets observed in the Gliese 876 system (  M G

= 0.334 ± 0.03 M Θ

) as given in the following table, Gliese System

Mass

Semi Major Axis (au)

Gliese 876 b Gliese 876  Gliese 876 d Gliese 876 e

.7!6 " #.##$!% & #.71$ " #.##' % & 6.8' " #.$# %⊕ 1$.6 " 1.7 %⊕

#.#8' " #.##### #.16 " #.####$ #.##8 " #.######1! #.''$' " #.##1'

where  M ⨁ is  M  J 

 M ⨀

 is mass of *+n,

 M J 

 M  J 

= 1.89813 × 10

is mass of &+piter (  M Θ

27

kg 

 M ⊕

kg 

), and

= 1.9891 × 10

mass of arth.where mass of *+n,

= 1.89813 × 10

27

30

kg 

, mass of &+piter,

= 5.97219 × 10

24

kg 

, mass of arth, . -ind, -ind, if any of the exoplanets of Gliese 876 system have a resonant resonant orbits orbits (their synodi period is a m+ltiple integer of one of the periods period s). . olani olani ativity ativity on /o whih whih has a synhro synhrono+s no+s orbital orbital period, period, was proposed proposed to be the the res es+l +ltt of a tida tidall heat heatin ing. g. 0es es+l +lta tant nt tida tidall for fore e is the the die dierrene ene in gravitational fore experiened by the near and far sides of a satellite orbiting a parent body. %eas+rements of the s+rfae distortion of /o via satellite radar altimeter mapping indiate indiate that the s+rfae rises and falls by +p to 1## m d+ring one2half orbit. 3nly the s+rfae layers will move by this amo+nt. /nterior layers within /o will move by a smaller amo+nt, th+s we ass+me that on average, the ent entire ire mass ass of /o is moved oved thr thro+gh +gh !# m. %ass ass of &+p &+piter iter and /o are are  M  J 

= 1.89813 × 10

27

kg 

m Io

= 8.931938 × 10

22

k g 

  an and , res respeti petively vely.. Given Given that the peri4ove peri4ove and apo4ove apo4ove distanes distanes are r  peri 5 $## $#### ## m and and r apo apo  5 $'$## m respetively, respetively, the orbital period 1!8!' s, and the radi+s of /o is R/o 5 181.6 m, al+late the average power  of   of this tidal heating of &+piters gravitational fore (1 +  x ) n ≈ 1 + nx on /o. Hint : yo+ an +se the following approximation  for small x .

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'. 3n 7 %ay #1! at #18$, the o+ltation of  the star 9/: 8'1 ( δ −¿ *gr) by the asteroid 18! &+lietta on the star 9/: 8'1 ( δ −¿ *gr) was observed from ;orob+d+r temple, it lasted for only 6. s. e of asteroid &+lietta, if the semi ma4or axis of &+lietta a =2.9914  a+, and at o+ltation, the distane of &+lietta to the *+n and the arth is and .1!6 a+, respetively.

3076

a+

$. $. 0eently, an observer is +sing a hypothetial far2infrared arth2si>ed telesope with wavelength in the range of # to 6$# µm. 9e =nds a stati and ne+tral s+permassive bla hole with a mass of 1 billions (.1 x 1#1#)  M ⨀ . ?etermine the maxim+m distane from the observer to this bla hole that an be meas+red with his telesope.@owadays, a s+permassive bla holes +p to 1 billion (.1 x 1#1#)  M ⨀ has been disovered. /n the 9ershel *pae 3bservatory, aed telesope with wavelength in the range of # to 6$# m. 9e =nds a s+permassive bla hole with a mass 1 billion

 M ⨀

. ?etermine the

maxim+m2observer distane to this bla hole that an be meas+red with his telesope.

!. 21 5 1#2' ergs s21 m2

9>21 1#... 

3 / 4

11.< mole+lar lo+d is nown to have a temperat+re T   5 11! . /f interstellar mole+les (ass+med spherial) have masses of abo+t ! H 1# 2' g g and radii of  '.! I and are in thermal eB+ilibri+m with the s+rro+nding gas, an yo+ estimate the freB+eny they will radiateE ?esribe yo+r answer mathematially. 1.Jonsider a star with whoseits mass density is inversely proportional to the radial distane from the enter of the star, with the fator of proportionality is ❑ ¿ 5.0 × ❑ kg / m . /f the esape veloity of any ob4ets to leave the star is 2

4

v 0 =1.5 × 10 m / s

, al+late the mass of the star.

(hintE) 1'.