What are the coordinates of the north star using cross product of given vectors?

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In summary, the conversation discusses a problem where the location of the north star needs to be determined on Earth. Three measurements are taken using a sextant and compass, and the spherical coordinates are converted into Euclidean coordinates. The first part of the problem involves finding the length of the sidereal day using dot product angle calculations. The second part involves finding the coordinates of the north star using the cross product, although there is some confusion about which vectors to use for the calculation. The final answer is given as (240,60) in spherical coordinates.
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ElijahRockers
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Homework Statement



Ok assume we're on earth, we don't know which way is north, and we don't know our latitude.

we can track a single star. We make three measurements (m1, m2, m3) with our sextant and our compass. The first measurement at time=0:00 hours, the third at exactly 1:00 hour, and the second roughly near 30 mins. I converted the spherical coords into euclidean coords:

M1: (.70711, 0, .70711) M2: (.74267,-.1155,.65962) M3: (.75861,-.21125,.61635)

The initial part of the question is to find the length of the sidereal day. I did that by using some dot product angle magic.

The second part, which is arguably much simpler, is giving me some trouble. We are supposed to find the coords of the north star. I know we're supposed to use the cross product, but I completely forgot what I am supposed to take the cross product of... I would assume it's the same vectors I used to calculate the length of the sidereal day, but I don't understand how that will give me the coords of polaris.

If someone could explain to me how the cross product of those vectors can yield the north star coords, I would greatly appreciate it, thanks!

The Attempt at a Solution



The answer (in spherical coords) is (240,60), if you assume M1 to lie on the x-axis. 240 is the longitude, and 60 is either the latitude or the colatitude. looking at the next parts of the question (which I know how to do) i am assuming the latitude is 60.
 
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I am still not sure how I can use the cross product of those 3 vectors to get the coords of the north star.
 

FAQ: What are the coordinates of the north star using cross product of given vectors?

What is the Celestial Sphere Problem?

The Celestial Sphere Problem is a concept in astronomy that refers to the apparent motion of the stars and other celestial objects as observed from Earth. It is a way of visualizing the sky as a giant dome surrounding the Earth, with all the stars and celestial objects fixed on its surface.

How does the Celestial Sphere relate to the Earth's rotation?

The Celestial Sphere is a convenient way of visualizing the Earth's rotation and the apparent movement of the stars. As the Earth rotates on its axis, the stars appear to move across the sky in a circular pattern. This is because the Earth's rotation causes us to see different parts of the Celestial Sphere at different times.

Is the Celestial Sphere a physical object?

No, the Celestial Sphere is not a physical object. It is a theoretical construct used by astronomers to understand and visualize the movements of celestial objects in the sky. It is a helpful tool in understanding the complex motions of the stars and other celestial bodies.

How does the Celestial Sphere help astronomers map the sky?

The Celestial Sphere provides a reference point for astronomers to map the positions of stars and other celestial objects in the sky. By imagining the sky as a giant dome surrounding the Earth, astronomers can use coordinates and angles to pinpoint the location of celestial objects in relation to each other.

Are there any limitations to using the Celestial Sphere?

Yes, there are some limitations to using the Celestial Sphere as a model for understanding the sky. It assumes that all celestial objects are fixed on the surface of the dome, which is not entirely accurate. It also does not take into account the actual distances between objects, as they may be at varying distances from Earth. However, it is still a useful tool for visualizing and understanding the movements of celestial objects in the sky.

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