On April 26,1997, astronomers announced the discovery of a...

On April $26,1997,$ astronomers announced the discovery of a new planet orbiting the star Rho Coronae Borealis (Sky and Telescope, July 1997). The astronomers deduced the existence of the planet by measuring the change in line-of-sight velocity of the star over a period of ten months. The measurements appear to fall along a sine wave, as shown in the accompanying figure. a. What are the period, amplitude, and equation for the sine wave? b. How many earth days does it take for the planet to orbit Rho Coronae Borealis? c. Use the equation to estimate the change in line-of-sight velocity on day 36 d. On day 36 do you think that the planet is between the earth and Rho or that Rho is between the earth and the planet? CAN'T COPY THE FIGURE

On April $26,1997,$ astronomers announced the discovery of a new planet orbiting the star Rho Coronae Borealis (Sky and Telescope, July 1997). The astronomers deduced the existence of the planet by measuring the change in line-of-sight velocity of the star over a period of ten months. The measurements appear to fall along a sine wave, as shown in the accompanying figure.
a. What are the period, amplitude, and equation for the sine wave?
b. How many earth days does it take for the planet to orbit Rho Coronae Borealis?
c. Use the equation to estimate the change in line-of-sight velocity on day 36
d. On day 36 do you think that the planet is between the earth and Rho or that Rho is between the earth and the planet?
CAN'T COPY THE FIGURE

Key Concepts

Sine Function Equation

The sine function equation combines the key parameters—amplitude, period, phase shift, and vertical shift—to provide a precise mathematical model of periodic data. This equation is essential for predicting values at different time points and for analyzing the behavior of systems that exhibit cyclic variations, such as the motion of stars under the influence of orbiting planets.

Amplitude

Amplitude refers to the maximum deviation of a sine wave from its central value. It represents the strength or intensity of the oscillation. In astronomical applications, such as measuring a star’s line-of-sight velocity variations, the amplitude indicates the maximum radial velocity change induced by the gravitational pull of an orbiting exoplanet.

Radial Velocity Method

The radial velocity method is a technique used in astronomy to detect exoplanets by measuring variations in the velocity of a star along the line of sight. These variations, often visualized as a sine wave, are caused by the gravitational pull of an orbiting planet causing the star to wobble. Analyzing this sinusoidal change enables astronomers to infer properties of the planet, including its orbital period and mass.

Sine Wave

A sine wave is a fundamental periodic function used in mathematics and science to model oscillatory behavior. It is characterized by parameters such as amplitude, period, phase shift, and (if necessary) a vertical shift, which together determine its shape and position on a graph. In many applications, including astronomy, sine waves effectively represent regular, repeating phenomena.

Period

The period is the duration required to complete one full cycle of a sine wave. It is a critical parameter that indicates the frequency of the oscillation. In the context of astronomical measurements, the period derived from a sine curve fitted to observational data can correspond to the orbital period of an exoplanet around its star.

Transcript

00:01 So the question tells us that there is a sun or a star called row, and there is a planet orbiting it.

00:10 So i'm going to draw a picture.

00:12 So this one will be the sun, you know, color it red.

00:20 And then we have the planet row orbiting around it like this.

00:26 And then we have earth over here.

00:30 I didn't type this up, but so here's earth.

00:35 And basically scientists were measuring the distance that the planet was compared to the sun like this, and it is modeled by the sinusoid over here.

00:50 So the first question asks, find the period, amplitude, and equation for the sine wave.

00:56 So i'll scroll down a little bit.

00:59 So the period is going to be how often it repeats.

01:03 And by looking at this, this is going to be one cycle.

01:06 This entire graph is one cycle.

01:09 So our period is going to be 40 days.

01:17 And the other section is asking for the amplitude.

01:24 Amplitude is going to be 65.

01:27 That's marked there.

01:27 So 65 meters per second.

01:36 And then lastly, it asks us to write the equation or the sine wave.

01:42 So we have y equals.

01:44 Our amplitude is 65 times sign since it's beginning on the midline at x equals 0 of period is 40 days so we're trying to look for this b value b times x minus c this b value the b value is going to so period is going to be equal to 2 pi over b i plugged in period for 40 already so i could solve this so 40b is equal to 2 pi, and therefore b is equal to pi over 20.

02:25 So we have sign of pi over 20, x, that we don't have a phase shift since it begins at the origin.

02:34 So that's it.

02:35 And we don't have a vertical shift.

02:36 So here is our equation.

02:40 And that's part a.

02:42 Next part b, it asks how many earth days does it take for the planet to orbit row? and for the planet you orbit once, it's basically the cycle of the planet going down this way and then going behind row and then coming back again.

03:03 And that's basically the same thing as one period.

03:06 As the planet is going down this way from this middle axis, the line of side velocity is increasing.

03:14 So this size plus, the size minus.

03:17 So as it goes up goes down this way, it increases to 65 and then it goes back again to its be when it's behind the sun.

03:28 And that's when it's going to be zero.

03:30 And then it's going to come out from the other side, which makes it a negative 65 and it will go back again.

03:36 So it's basically asking for the period, which we found here is 40 days.

03:41 So this is for part a.

03:43 Part b would be 40 earth days.

03:49 Since that is one cycle in the graph.

03:52 Now for part c, it says estimate the change in line of sight velocity.

04:00 This is the change of line in sight velocity, this y -axis.