Skip to main content

Bernard's Star is 5.963 light years away with a proper motion of 10.33 arc seconds per year and a hydrogen alpha line measured at 656.047 nm. What...

There are two components of the motion here: The proper motion, which we see as horizontal because it is moves the star neither closer nor further relative to us, and the redshift, which is due to the expansion of the universe and also the star's inherent movement toward or away from us. These are two perpendicular components of a vector that comprises the star's velocity relative to our solar system.

Let's start with the proper motion. For a star at distance `r` away from us with a proper motion in radians of `{d theta} / {d t}`the horizontal velocity is:
`v = d/dt[r tan theta] = r sec^2 theta {d theta}/{d t} approx r {d theta}/{d t}`
Since the angles are so small (arcseconds), we don't actually need to worry about the secant term, which is very close to 1.


The important thing is to convert to radians; there are 3600 arcseconds in a degree, and `180/pi` degrees in a radian.


`10.33 {arcsec}/{yr} * ({1 deg}/{3600 arcsec})({pi rad}/{180 deg}) = 5.01*10^-5 {rad}/{yr}`
Then just multiply by the distance:
`v = r {d theta}/{dt} = (5.963 light-yr)(5.01*10^-5 {rad}/{yr}) = 2.97*10^-4 {light-yr}/{yr}`
Now we need to figure out the redshift. The hydrogen alpha line has redshifted to 656.047 nm, but what was it originally? We actually need quite a precise answer in this case. The most precise value I've been able to look up is 656.281 nm. Our the observed wavelength is shorter, indicating blueshift; so this star is actually moving toward us, against the expansion of the universe.

To find its velocity, we use the Doppler shift formula:
`lambda_1/lambda_0 = sqrt{{v+c}/{v-c}}`
For velocities this small, we can use an approximation instead:
`v/c approx {lambda_1 - lambda_0}/{lambda_0}`
Where a positive value for `v` indicates redshift, and a negative value indicates blueshift. Substitute in, recalling that by definition `c = 1 {light-yr}/{yr}` :
`v = c {656.047 nm - 656.281 nm}/{656.281 nm} = -3.57*10^-4 {light-yr}/{yr}`


Those are each of our components. To get our overall velocity and angle of approach, we just use the standard formulas for vector components:
`v = sqrt{v_x^2 + v_y^2} = sqrt{(2.97*10^-4)^2 + (-3.57*10^-4)^2} = 4.64*10^-4 {light-yr}/{yr}`
Call the angle `phi` ; note that this angle is not negligibly small:
`phi = tan^{-1} (v_y / v_x) = tan^{-1} ({2.97*10^-4}/{3.57*10^-4}) = tan^{-1} (0.832) = 39.8 deg ` This star is approaching us at a speed of 4.64*10^-4 light-years per year, at an angle of 39.8 degrees from head-on. Since it is 5.963 light-years away, if nothing else changed, it would go past us in a little over 10,000 years.

Comments

Post a Comment

Popular posts from this blog

Is there a word/phrase for "unperformant"?

As a software engineer, I need to sometimes describe a piece of code as something that lacks performance or was not written with performance in mind. Example: This kind of coding style leads to unmaintainable and unperformant code. Based on my Google searches, this isn't a real word. What is the correct way to describe this? EDIT My usage of "performance" here is in regard to speed and efficiency. For example, the better the performance of code the faster the application runs. My question and example target the negative definition, which is in reference to preventing inefficient coding practices. Answer This kind of coding style leads to unmaintainable and unperformant code. In my opinion, reads more easily as: This coding style leads to unmaintainable and poorly performing code. The key to well-written documentation and reports lies in ease of understanding. Adding poorly understood words such as performant decreases that ease. In addressing the use of such a poorly ...
Post a Comment
Read more

A man has a garden measuring 84 meters by 56 meters. He divides it into the minimum number of square plots. What is the length of the square plots?

We wish to divide this man's garden into the minimum number of square plots possible. A square has all four sides with the same length.Our garden is a rectangle, so the answer is clearly not 1 square plot. If we choose the wrong length for our squares, we may end up with missing holes or we may not be able to fit our squares inside the garden. So we have 84 meters in one direction and 56 meters in the other direction. When we start dividing the garden in square plots, we are "filling" those lengths in their respective directions. At each direction, there must be an integer number of squares (otherwise, we get holes or we leave the garden), so that all the square plots fill up the garden nicely. Thus, our job here is to find the greatest common divisor of 84 and 56. For this, we prime factor both of them: `56 = 2*2*2*7` `84 = 2*2*3*7` We can see that the prime factors and multiplicities in common are `2*2*7 = 28` . This is the desired length of the square plots. If you wi...
Post a Comment
Read more

What warning does Chuchundra issue to Rikki?

Chuchundra, the sniveling, fearful muskrat who creeps around walls because he is too terrified to go into the center of a room, meets Rikki in the middle of the night. He insults Rikki by begging him not to kill him. He then insults him by suggesting that Nag might mistake Chuchundra for Rikki. He says, "Those who kill snakes get killed by snakes."  He issues this warning to Rikki not to help keep Rikki safe but as a way of explaining why Rikki's presence gives him, Chuchundra, more reason to fear.  Chuchundra starts to tell Rikki what Chua the rat told him--but breaks it off when he realizes he might be overheard by Nag. He says, "Nag is everywhere, Rikki-Tikki." Rikki threatens to bite Chuchundra to get him to talk. Even then, Chuchundra won't overtly reveal any information. But he does say, "Can't you hear, Rikki-Tikki?" This is enough of a clue for the clever mongoose. He listens carefully and can just make out the "faintest scratch-s...
Post a Comment
Read more