Assignment #8 Solutions - Astronomy 1 (1998)

Chaisson and McMillan (3rd Ed)
Chapter 6 Probs # 2, 9, 10
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#2
Compare your weight on Earth, Mars, Ceres and Jupiter
The force of gravity is given by Newton's gravitational Law:
		Weight = F = G m M/R2
	where M = mass of the planet, R = Radius of the planet
	G and m (your mass) are the same for all the cases

Form a ratio:   Wplanet/WEarth =  (Mplanet/MEarth)(REarth/Rplanet)2

The values of R and M from Table 6.1

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#9 
Sketch the path of Mars Pathfinder on its trip from Earth to Mars:
	Dec 4, 1996	Launched from Earth
	Mar    1997	Mars and Earth at Closest Approach (opposition)
	Jun  , 1997	Arrives at Mars

Note that the launch was 3 month before opposition and 
arrival was 3 months after opposition. Mars' orbital period 
is 1.88 times that of Earth and while the Earth moves 90 deg
in its orbit in 3 month, Mars will only move 48 deg in its orbit.


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#10
Find the time for a radio signal to travel to Saturn and back when
it is closest to Earth.
How far will a spacecraft orbiting Saturn at 100,000 km travel in 
that time?

Saturn is 9.5 AU from the Sun
Earth is 1.0 AU from the Sun
	Distance to Saturn = 8.5 AU = 8.5 x 150 million km
		= 1.28x109 km
	Time of travel = 2 (1.28x109 km)/300,000 km/s
		= 8530 seconds = 2.37 hours

What would be the speed of the spacecraft in its orbit?
 	Circular orbit speed = (GM/R)1/2
		G = 6.67x10-11 N m2/kg2
		M(Saturn) = 95 M(Earth) = 95 x 6x1024 kg
			= 5.7x1026 kg
So...
	v = (6.67x10-11x 5.7x1026/100,000,000 m)1/2
	  = 19,500 m/s = 19.5 km/s

	The spacecraft will travel  
	d = v t = 19.5 km/s (8500) s = 166,000 km
	This is 95 degrees around its orbit

	There is no way that the craft can be controlled in real
	time and control it adequately. The actions must be programmed 
	ahead of time by at least 2.4 hours

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Chapter 7 Probs # 2, 3, 4, 10
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#2
  Determine the surface gravity and escape velocity of the Earth for the case
if the average density were 3000 kg/m3 instead of 5500 kg/m3.

The mass of the planet is proportional to its density so...
M/Mearth = 3000/5500 = 0.545

The surface gravity is proportional to the mass of the planet so...
g/gearth = M/Mearth = 0.545
g = 0.545(9.8 m/s2) = 5.35 m/s2

The escape velocity is proportional to the square root of the planet mass.
	vesc = (2GM/r)1/2
	
	vesc/vearth = (M/Mearth)1/2
		= (0.545)1/2 = 0.74
So...
	vesc = 11.2 km/s (0.74) = 8.3 km/s

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#3
Find the mass of the Earth's atmosphere
assuming:
	density is uniform = 1.3 kg/m3
	thickness = 7.5 km
	Radius of the Earth = 6371 km

The atmosphere is like a coat of paint on the surface of a sphere. The volume
is the thickness times the area of the sphere.
	Area of a sphere = 4(pi)r2
		= 4(3.14)(6371 km)2 = 5.10x108 km2
	Volume = Area x thickness = 5.10x108 km2 x 7.5 km
		= 3.83x109 km3
	1 km3 = 109 m3
	volume	= 3.83x1018 m3
	Mass = density x volume = 1.3 kg/m3 x 3.83x1018 m3
		= 5.0x1018 kg
	The Mass of the Earth is 6x1024 kg
	Mass of Atmosphere/Mass of Earth = 5.0x1018 kg/6x1024 kg
		= 8.3x10-7
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#4
Greenhouse effect on Earth:
	With Greenhouse effect Tearth = T1= 250 K + 40 K = 290 K 
	Without Greenhouse effect Tearth = T2= 250 K

	Determine the fraction of radiation captured by the greenhouse gases.

	Use Stefan's Law: The flux = s T4
	The trapped radiation is  
		F1 - F2 = s (T14 - T24)
	The fraction is [F1 - F2]/F1 = 1- F2/F1 
		= 1 - (T2/T1)4
		= 1 - (250/290)4 = 0.45
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#10
Compare the tidal force on the Earth due to Jupiter with that due to the Moon.
Assume Jupiter is 4.2 AU from the Earth.

	Tidal forces are proportional to  M/r3
	where M = mass of planet causing the force and 
	r = the distance to the planet
		rMoon = 384,000 km = 0.00256 AU

	Form a ratio of FJupiter/FMoon 
		= (MJ/MM)(rM/rJ)3
		= (0.012/318)(0.00256/4.2)3
		= 8.5x10-15

	This is such a small quantity that no alignment of the planets which 
	have masses smaller than Jupiter are going to have any effect on the Earth!!
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