Exercise 22.1.1: Dimensional analysis in .NET Incoporate Code 128C in .NET Exercise 22.1.1: Dimensional analysis

Exercise 22.1.1: Dimensional analysis using visual .net todraw code 128 in web,windows application RM4SCC Fill in the code 128 barcode for .NET missing steps above that show that the dimensions of energy ux are kg s-3 . Then show similarly that the dimensions of c 3 /G times the square of the frequency are the same.

. Exercise 22.1.2: Size of gravitational wave ux We saw that code 128 barcode for .NET a gravitational wave arriving at the Earth might have an amplitude h as large as 3 10-21 . If its frequency is 1000 Hz, then calculate the energy ux from such a wave.

Compare this with the ux of energy in the light reaching us from a full Moon, 1.5 10-3 W m-2 . Use Equation 9.

2 on page 108 to compute the apparent magnitude of the source. Naturally, the source is not visible in light, so this magnitude does not mean a telescope could see it, but it gives an idea of how much energy is transported by the wave, compared to the energy we receive from other astronomical objects..

a If you ar e puzzled by the idea of a negative frequency, remember that frequency is the number of cycles of the wave per unit time. If we run time backwards, such as by making a lm of the wave and running it backwards, then the number of cycles per unit time also goes backwards, and the wave has a negative frequency. But the backwards-running lm shows a normal wave, one that you could have created in the forward direction of time with the right starting conditions, so it must also have a positive energy.

. light-years Code 128 Code Set A for .NET , or r = 4.6 1023 m, we get h 6 10-21 .

Our argument gives this as an upper bound on the strength of waves from such a source, and therefore on the distortions in shape that the wave produces in a detector. How far below this upper bound do realistic wave amplitudes lie Clearly this depends on the source. But when motions are not highly relativistic, it is possible in general relativity to make a simple approximation that works very well.

The source must be the mass of the system radiating, since both the active gravitational mass and the active curvature mass are dominated by the ordinary mass-energy. But the overall mass of the system is constant and gives rise to the spherical Newtonian eld, not to waves. We are looking for the part of the mass-energy that can follow.

This upper limit on realistic gravitational waves has set a target for detector developers since the 1960s.. 22. Gravitational waves the pattern s in Figure 22.1 on page 312. It should not be surprising, therefore, that in general relativity: gravitational radiation is produced only by the mass-equivalent of that part of the kinetic energy of the source that has the elliptical pattern of Figure 22.

1 on page 312 as seen from the direction of the observer of the gravitational radiation. Written as an equation, the prediction of general relativity is called the quadrupole formula. It is similar to the expression for the corrections to the coef cients of the interval that we computed in 18: h= 8G rc2 K c2 .

. projected elliptical part The notatio n projected elliptical part here means that only the part of the kinetic energy that contributes to source motions similar to those of the test particles in Figure 22.1 on page 312 contributes to the radiation. Each polarization must be treated separately.

The factor of eight is not something we can derive here; we must just accept that a full calculation in general relativity justi es it. It takes into account both the mass-energy and pressure parts of the source as well as any gravitational potential energy (the non-linearity of Einstein s equations)..

(22.3). This gives a good approximation for radiation from systems where the velocities are small compared to c. Einstein was the rst to derive the quadrupole formula and yet, as I remarked earlier, he did not always have con dence in it. It took decades for physicists to be sure that it represented a good approximation, especially for realistic systems where gravitational potential energy was comparable with the kinetic energy.

There were important contributions from Landau (whom we met in 20) and his Soviet colleague Yvgeny Lifshitz (1915 1985), and from Chandrasekhar (see 12), among many others. The subject is still an important area of research today, though not a controversial one. Physicists are developing better and better approximations to the radiation by re ning Equation 22.

3, in order to be able to recognize and interpret gravitational waves in the observations made by the detectors that we will describe below.. In this sec .net framework Code 128 Code Set C tion: with the help of an analysis we calcualte the energy carried by a gravitational wave. We see that even weak waves carry huge energies.

. We introduc ed the idea of energy ux in 9, where we discussed the apparent brightness of stars. The apparent magnitude of a star is a measure of the ux of light energy we receive from it. By analogy, we have here the formula for the energy ux carried by a gravitational wave.

. Gravitation VS .NET Code128 al waves carry energy, lots of energy Gravitational waves clearly can transfer energy from one system to another. For example, if the particles in Figure 22.

1 on page 312 are embedded in a viscous uid, then their motion will transfer energy to the uid, and long after the wave is gone the energy will remain. The energy transferred should be small, because we know the waves have great penetrating power. To nd out what energy is carried by the waves requires a small calculation, so it is reserved to Investigation 22.

1 on the preceding page. The result, however, is important enough to write down here. Let us consider a plane wave.

This is a wave from a source that is so far away that the wave passes us with a at wave front, all parts of the wave traveling in the same direction with the same amplitude h. Suppose in addition that the gravitational wave is a simple sine-wave oscillation with a frequency f (measured in hertz). The appropriate measure of the energy carried by the wave is its energy ux, the energy carried by the wave through a unit area per unit time.

The formula derived in Investigation 22.1 on the previous page is c3 2 2 F= f h . (22.

4) 4G The key point about this formula is that the energy is proportional to the squares of the amplitude and of the frequency. Each of the two polarizations of the wave contributes its own energy, so this formula must be used separately for the + and amplitudes. The constant c3 /G is a very large number, so that even when h is as small as we have found it to be, the ux can be large.

In Exercise 22.1.2 on the previous page.

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