Frank D. Drake, an investigator in the SETI (Search for...

Frank D. Drake, an investigator in the SETI (Search for Extra-Terrestrial Intelligence) program, once said that the large radio telescope in Arecibo, Puerto Rico, "can detect a signal which lays down on the entire surface of the earth a power of only one picowatt." (a) What is the power that would be received by the Arecibo antenna for such a signal? The antenna diameter is $300 \mathrm{~m} .$ (b) What would be the power of a source at the center of our galaxy that could provide such a signal? The galactic center is $2.2 \times$ $10^{4}$ ly away. Take the source as radiating uniformly in all directions.

Key Concepts

Astronomical Distance Units

Astronomical distance units, such as light years, are used to express distances on cosmic scales. These units are essential in calculations involving the propagation of electromagnetic radiation from distant sources, as they provide a means to convert between everyday units and the vast distances encountered in astronomy. Accurate conversion between these units allows for the proper application of physical laws like the inverse square law in astrophysical contexts.

Isotropic Radiation

Isotropic radiation is the idea that a source emits energy uniformly in all directions. This assumption is often used in astrophysics to simplify calculations involving the power distribution from a cosmic source. When a source radiates isotropically, the total emitted power is spread evenly over the surface area of a sphere centered on the source, which grows with the square of the distance from the source.

Antenna Aperture

The concept of antenna aperture refers to the effective area of a radio telescope or antenna that is sensitive to incoming electromagnetic waves. It dictates how much of the signal’s power is captured by the antenna. For a given dish diameter, the aperture is used to compute the physical area available to intercept the wave, which directly influences the received power from a signal of a particular power density.

Inverse Square Law

The inverse square law is a fundamental principle used to describe how a physical quantity or intensity, such as electromagnetic radiation, diminishes with increasing distance from its source. It states that the observed power per unit area is inversely proportional to the square of the distance from the source. This principle is crucial when calculating how much of a signal’s power is received at different distances, as is done when estimating the power at Earth from a distant radiating source.

Transcript

00:01 So in this question we're asked about the arcebo telescope and we're asked to determine what would the power of a signal be that was received by the arcebo antenna if the signal which could which if the signal could lay down on the entire surface of the earth a power of only one peak of what.

00:22 Then in part b we're asked what would the power of a source at the center of the galaxy be for such such as? a signal given that the galactic center is 2 .2 by 10 to the four light years away.

00:37 So and we're also told that the arrasibo telescope is 300 meters in diameter.

00:42 So that's 150 meters in radius.

00:45 So the most important idea of this question is, so for a, let's start with a, is that the intensity received by the aricebo telescope, so ia, is equal to the intensity received by the intensity of the by the intensity of the source over the surface of the earth.

01:06 So i .e.

01:08 That's intensity over the surface of the earth.

01:11 And ia is the intensity over the telescope.

01:15 So we know that the intensity over the telescope, where we know that intensity in general is equal to the power over the area.

01:23 So that means that the intensity over the telescope, so ia is equal to, so the power of which we want to find out.

01:32 So p over the area.

01:33 Of the telescope so that's pi by or a squared since we're going to take it to be a circle and this is equal to that so it's pi of r a squared and this is equal to the area to the intensity of that the earth receives and we're told that it receives that the signal produces one one by 10 to the minus 12 pica watts per over the surface of the earth so that's going to be so it's 1 by 10 to the minus 12 watts so that's our power in this case and then over the surface of the earth so that's surface of a sphere surface area of the sphere so that's 4 pi and or the radius of the earth is 6 .37 so it's ore e squared so then if we rearrange just get an expression for p so we get that p is equal to 1 1 by 10 to minus 12 watts by so the pies cancel each other at so it's going to be the radius of the arcebo telescope, so that's squared.

02:47 So that's 150 meters squared over four by the radius of the earth squared.

02:56 So 6 .37 by 10 to the 6 squared.

03:00 And if we work this out, we get that the power, the power of the signal of the earth is that are at the, the power of the signal that the arcebo telescope receives is equal to 1...

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