Position and velocity vector of point referred to equinox

[solarblast]

I'm looking at a book on meteor calculations involving at two Earth stations, A and B. The computations for A are given, and the same for B. I would like to know a reference for equations 12, 13, and 14. Even a derivation.

I find figure 4 a bit puzzling. The text uses lower case phi, a vertical line with a circle in the middle, but the figure shows what is perhaps a capital phi. A looping curve. I'm guessing this is meant to be former symbol, vertical + circle,

Secondly, it looks like the x,y,z axes for the topo system should be labeled with a prime on x,y,z.

Thirdly, the two phi's look like they should have the primes swapped.

Comments on my remarks?

 

[cepheid]

The symbols \phi and \varphi are just two different ways of writing lowercase phi.

For a derivation of equation 12, see this diagram here:

http://en.wikipedia.org/wiki/File:3D_Spherical.svg

How to get the Cartesian coordinates (x,y,z) of the point in terms of its spherical coordinates (r,θ,ϕ)?

It's clear from the geometry that the z-coordinate of the point is just given by z = rcosθ. (Just take the position vector of the point and project it onto the z-axis). EDIT: or just look at the right triangle formed by the dashed lines.

The magnitude of the projection of the position vector onto the xy-plane is just rsinθ. It's the other side of the (dashed) right triangle that we used above to get the component of the vector that was parallel to the z-axis. So it's the dashed line on the xy-plane in the figure.

Now, take this *projected* vector and further resolve IT into an x-component and a y-component on the plane. The x-component will be given by the projected vector multiplied by the cosine of the angle between that vector and the x-axis:

x = (rsinθ)cosϕ

Similarly:

y = (rsinθ)sinϕ

Again, you get these by drawing the appropriate right triangle on the plane (not shown).

EDIT: The meanings of theta and phi are SWITCHED in my figure compared to yours. Sorry for any confusion. Mathematicians and physicists often use the opposite symbols for these two angles.

 

[cepheid]

I'm looking at a book on meteor calculations involving at two Earth stations, A and B. The computations for A are given, and the same for B. I would like to know a reference for equations 12, 13, and 14. Even a derivation.

I gave you the derivation for 12 in my previous post. Equation 13 is just obtained by differentiating equation 12 with respect to time, and using the chain rule. The equation makes perfect sense if you assume that \phi_A^\prime is constant i.e. it does not vary with time. This makes sense since it is supposed to be the latitude of the station, which is fixed.

Equation 14 is just the same as equation 12, except in the station's coordinate system. Also, every component has been divided by the total length of the vector ("R") since this is meant to be a UNIT vector in the direction of the zenith. That's why 14 looks like 12, but without the radius as a coefficient on each term.

Secondly, it looks like the x,y,z axes for the topo system should be labeled with a prime on x,y,z.

I'm sorry, I don't know what you mean here.

Thirdly, the two phi's look like they should have the primes swapped.

Not from what I can see in the text. It says that \phi_A^\prime is the latitude of station A as measured relative to the centre of the Earth, and that is reflected in the diagram.

 

[solarblast]

Thanks very much. That was almost too easy now that I look at it, but what threw me off was the primes, and the fact that ϕ′A was a constant. That plus mention of the word equinox, which suggested some deeper meaning than I could see.

I think the use of primes was different than I expected. When I used topo, I was thinking the coordinate system at the surface of the Earth should have been marked with a prime (x',y',z'). I think the case here is the author used ϕ′A to distinguish it from ϕA.

Regarding (14), then this seems to me that is the same transformation applied as (12), as you say, to the topo (station) system, but ϕA' is replaced by ϕA. It is further modified to make it a unit vector. In other words, the transform is applicable to either system.
Somewhere on this thread I believe there's a math tool that allows one to select Greek letters and maybe math symbols. I don't see it at the moment. Ah, it's the Advanced button.

Bonus question. This pub was written in the sixties, and it seems to me the author slips into using vector when speed is meant. This may be a habit of the times. Velocity to me means vector and not a scalar (magnitue). See the attachements from a meteor pub in 1925. Whoops, there is no way to attach them here, so I'll post them next. Maybe. It may be that only one set applies to a thread.

 

[cepheid]

Thanks very much. That was almost too easy now that I look at it, but what threw me off was the primes, and the fact that ϕ′A was a constant. That plus mention of the word equinox, which suggested some deeper meaning than I could see.

I think that "equinox of date" might just be referring to the reference meridian for the measurement of sidereal time (i.e. the great circle for which θ is considered to be 0). So, the position θ is measured relative to this point. However, I encourage you to look into it further rather than taking my word for it.

I think the use of primes was different than I expected. When I used topo, I was thinking the coordinate system at the surface of the Earth should have been marked with a prime (x',y',z'). I think the case here is the author used ϕ′A to distinguish it from ϕA.

Okay, sure. What is topo, by the way?

Regarding (14), then this seems to me that is the same transformation applied as (12), as you say, to the topo (station) system, but ϕA' is replaced by ϕA. It is further modified to make it a unit vector. In other words, the transform is applicable to either system.

Yes, of course it's applicable in both cases. It's a general transformation between a spherical and a related Cartesian coordinate system. The way the two systems are related is that the z-axis is a line passing between the two poles of the spherical system, and the x and y axes lie in the equatorial plane of the spherical system.

Somewhere on this thread I believe there's a math tool that allows one to select Greek letters and maybe math symbols. I don't see it at the moment. Ah, it's the Advanced button.

Yes, to generate mathematical formulae, the forum uses LaTeX, which is a typesetting system used in publishing professional scientific and mathematical publications. You can use the button, but you can also right click on my phi symbols and select "show TeX commands" from the menu that pops up in order to see the LaTeX code that was used to generate it. There is also a LaTeX tutorial here: https://www.physicsforums.com/showthread.php?t=546968

Bonus question. This pub was written in the sixties, and it seems to me the author slips into using vector when speed is meant. This may be a habit of the times. Velocity to me means vector and not a scalar (magnitue). See the attachements from a meteor pub in 1925. Whoops, there is no way to attach them here, so I'll post them next. Maybe. It may be that only one set applies to a thread.

Can you give me an example of where you think the term vector is misused? I can only see legitimate uses of the term vector. In particular, the x, y, and z components of the position and velocity vectors are given. If they weren't vectors, they wouldn't need to be described using components.

 

[solarblast]

The vector/velocity/magnitude example is in a book by Oliver, 1925, on meteors. I cannot figure out a way to show you the page. I have it on a jpg file. As far as I can tell from the pdf of the book, he never uses the word vector. If you are familiar with American baseball, physicists often are often critical when a game announcer says, "The pitcher has a lot of velocity on the ball.", instead of speed.

I'll look further into "equinox date".

I believe topo means on the surface of the earth. So a topocentric coordinate system might be az/el. From wikipedia,"Geocentric coordinates are an Earth-centered system of locating objects in the solar system in three-dimensions along the Cartesian X, Y and Z axes. They are differentiated from topocentric coordinates which use the observer's location as the reference point for bearings in altitude and azimuth. "

 

[solarblast]

See the attachment from Oliver's Meteor book. Pages 166 and 167 in particular. Those mentions of velocity look more like magnitudes.

 

[cepheid]

It could be that this is a 1D problem and the velocity can be represented as a signed scalar, with the sign indicating direction.

Or it could be that the author is using the term "velocity" and then writing down an expression for the magnitude of the velocity. My response would be: who cares? At the level the text is intended for, everyone knows that velocity is a vector and nothing is to be gained by splitting hairs or being too pedantic.