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Astrophysics and Cosmology

I. Stars and Distances

Note: References to the Giancoli text are in the form G:33-6, where the 6 means the sixth section.

Study the picture at the left and read G:33-1.
tanf=d/D and f=90-q
and
D=d/tanf
where d is the radius of the Earth's orbit, 1.5x10¹¹ m, also known as the Astrnomical Unit (AU). One parsec (pc) is the distance to a star with a paralax angle f equal to 1 second of arc (1/3600 degree). Thus a parsec equals
1.5x10¹¹ m/tan(1/3600) = 3.1x10¹⁶ m = 3.26 ly = 206265 AU

Interactive Example

If q equals 89.9999( degrees, what is f in seconds? Enter your answer below and click outside.


What is the distance ito the nearest tenth of a parsec? Enter your answer below and click outside.


What is the distance in light-years (ly) to the nearest tenth? Enter your answer below and click outside.

Another way to estimate the distance to a star is to use both the Luminosity, L, (The total power in watts emitted by a star) and the apparent brightnesss, l, (watts/m² at the earth). Suppose you have reason to believe that you know the Luminosity of a star. (Read Giancoli 33-2 to learn how this is possible.) You can use this to find the distance to the star since

l = L/4pR².

Interactive Example

The sun transmits 1350 watts/m² to the earth's orbit and has an absolute luminosity of about 3.8x10²⁶ watts, what is the distance to the sun to the nearest Gm(10⁹ m)? Enter your answer below and click outside.

More information can be learned from the star by studing the relationship between the wavelengths of the light it emits and the magnitude of the light it emits at that wavelength. According to Wien's Law, the product of the wavelength, l_max , at which the peak radiation occurs and the temperature, T, of the star is 2.90x10^-3 mK.

Interactive Example

If the Sun's spectrum peaks at 500 nm(10^-9 m), what is its temperature?

The Hertzsprung-Russell (H-R) Diagram

Interactive Activity

Stars can be located on a H-R diagram shown on the left. From what you learned about our Sun first change the pull down menu below to "Sun" and then click the location of the Sun on the diagram. Assuming you have read Giancolli 33-2, Change the pull down menu to 'Red Giant' and click on the location of the Red Giants. Next do the same for the 'White Dwarfs'. Choose SUN Red Giant White Dwarf

Hubble's Law

Interactive Activity

To the left is a representation of the visible light spectrum , from Bright red (l = 750 nm) to violet (l = 400 nm). The vertical black line at the near end of the violet region is the 410 nm spectral line of atomic Hydrogen. Depending on the speed of the Galaxay this spectral line will be shifted to the left (red shifted) when viewed from the earth. Click on the spectrum to the left of 410 nm and you will find the corresponding recession velocity and distance to the galaxy, as well as the wavelength you chose here:

Wavelength (nm)	v/c	Distance (Mly)

Answer the following questions:

• What is the maximum velocity to the nearest hundredth of v/c of a galaxy can have and still have the 410 line fall in the visible range? Enter your answer below and click outside.
  
  
  
• What is the recession velocity if the distance to the galaxy is 2325 Mly? Enter your answer below and click outside.
  
  
  

Verify your results by using the tool and the formulae above.

II. General Relativity and the Theory of Gravity.

General Relativity is derivative from the Principle of Equivalence:

No experiment can distinguish between gravity and acceleration in any local (read small) region of space and time.

The picture on the left shows two balls falling in a room at the surface of the earth. On the right is a rocket accelerating at g with respect to the distant stars. Two balls fall inside the rocket just as they would on earth.



Study the picture at the left. An observer in an accelerating reference frame will see light curve as in the picture. This led Einstein to conclude that gravity must also bend light. Experiments confirmed this prediction.

Measuring distances between points in space when gravity is present is problematical. Any ruler used will bend because of the gravity. For that reason (and others) surveyors use light to measure such distances, but as we just learned light itself is bent by gravity. Thus the shortest distance we can measure in the presence of gravity appears to be a curved line!


Airlines regularly advertise that they "Fly the Great Circle Route to Tokyo (or wherever)". What this means is the route is the shortest distance between the departure and arrival. On a sphere the shortest distance betweentwo points is along the arc of a great circle on which the two points lie. A Great Circle is a circle whose center is at the center of the sphere. Lines of longitude and the equator (as shown in the figure) are examples of great circles, other lines of latitude are not great circles. Seen on a flat map the Great Circle route appears to be a curved line! A straight line drawn between the same two point on the map would actually correspond to a longer route. Whenever the shortest distance between two points is a curve, this is an indication that the underlying space is curved. This is the situation we find with light and gravity. The shortest distance between two points in the space surrounding a gravitating body as measured by light is curved. Thus Einstein draws the conclusion that our space is a curved surface of higher dimensional object. We can not directly perceive this higher dimension but we can detect its presence.

Imagine the existence of two dimensional creatures living on the surface of a three dimensional sphere such as the one in the picture. There are any number of experiments that they can do decide whether their space is curved or flat. For one, in flat space parallel lines never cross but in curved space they can and, in particular, on the surface of a sphere they always cross. Parallel lines can be constructed by drawing two lines perpendicular to a third line. On a sphere the lines of longitude are all perpendicular to the equator so they are all parallel and they all cross at the equator. Two of our two dimensional creatures starting North along parallel lines will find themselves being drawn together. They can either conclude that their space is curved or that there is a mysterious forcc pulling them to the North Pole. If they are Einsteinians they will conclude the former, if they are Newtonians they will think the latter.


When we discussed the Principle of Equivalence we emphasized the word local, that is because if the region of space is large enough the difference between gravity and acceleration can be detected as shown in the figure at the left. The two falling balls are drawn together as they fall towards the center of the earth. Newtonians would conclude that the earth was the source of a mysterious force called gravity, while we know better, we know that the mass of the earth curves the space around it and the 'falling ' bodies are following straight lines through this curved space (actually curved spacetime).

The most spectacular and best known consequence of this phenomenon is the prediction of the existence of Black Holes. Certain stars can collapse in such a manner that nothing can stop the collapse. They collapase them out of existence at least in this Universe. On their way to oblivion they reach a radius, called the Schwarzschild Radius, at which
R = 2GM/c².
At this radius, as viewed from outside, time stops and the gravity is so intense that nothing, not even light, can escape it. The Hole exists for eternity, but like the smile of the Chesire Cat in "Alice in Wonderland", the star that created it is no longer in existance. In the following SM refers to our Sun's Solar Mass. Choose a mass and the associated radius as well as the density will be calculated. The radius of the Sun is 7x10⁵ km and its density is 1175 kg/m³

Mass (kg)	Radius (km)	Density (kg/m3)
Choose SM 100SM 1000SM 10000SM 100000SM

© 1999 Carl Adler mailto:Carl@Image-ination.com