From polar coordinates to heliocentric ecliptic coordinates

In summary, the conversation discusses the conversion of polar coordinates to heliocentric ecliptic coordinates. The process involves converting to cartesian coordinates and then rotating the plane of reference. The specific steps and equations involved are also explained, with a focus on the effect of inclination on the position of the planet.
  • #1
Shukie
95
0
So I've calculated the polar coordinates of a planet, with the sun at the origin and the x-axis being the striped line going from the sun towards point P.

figure6.gif


Now I have to convert these polar coordinates to heliocentric ecliptic coordinates. To do this, I have to convert to cartesian coordinates first and then rotate the plane of reference so that the x-axis will point towards [tex]\Upsilon[/tex]. This is the answer:

figure7.gif


Converting to cartesian coordinates is easy, but then I'm lost. Could anyone tell me how exactly I go from [tex]x = r \cdot \cos{v}[/tex] to [tex](6)[/tex]?
 
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  • #2
It is easier to see what's going on if you first look at the situation when the inclination ##i## is zero and then rotate the plane of the orbit away from the ecliptic. This is shown in the figure below. We have auxiliary axes ##p## from the sun to the perihelion and ##q## perpendicular to ##p##. The heliocentric axes are ##x## and ##y##.
Heliocentric.png

We write unit vector relations
$$\begin{align} & \mathbf{\hat q}=\cos\Omega~\mathbf{\hat x}+\sin\Omega~\mathbf{\hat y}\\ & \mathbf{\hat p}=-\sin\Omega~\mathbf{\hat x}+\cos\Omega~\mathbf{\hat y}\end{align}$$The position of the planet is $$\begin{align} \mathbf{r}=r\cos(\omega+v)~\mathbf{\hat q}+r\sin(\omega+v)~\mathbf{\hat p}.\end{align}$$We now consider how these vectors change when the plane of the planet's orbit is rotated away from the ecliptic about the ##q##-axis to inclination angle ##i##. Only unit vecor ##\mathbf{p}## will change form. It will be off the plane of the ecliptic. Noting that its projection on the ecliptic is along its old direction. We have$$\begin{align}\mathbf{\hat p'}=\cos i (-\sin\Omega~\mathbf{\hat x}+\cos\Omega~\mathbf{\hat y})+\sin i~\mathbf{\hat z}.\end{align}$$We obtain the position of the planet in heliocentric coordinates in the inclined plane using equation (3) in which ##\mathbf{\hat p}## is replaced with ##\mathbf{\hat p'}## from equation (4) and ##\mathbf{\hat q}## from equation (1). We get $$ \mathbf{r}=r\cos(\omega+v)~( \cos\Omega~\mathbf{\hat x}+\sin\Omega~\mathbf{\hat y} )+r\sin(\omega+v)~cos i [(-\sin\Omega~\mathbf{\hat x}+\cos\Omega~\mathbf{\hat y})+\sin i~\mathbf{\hat z}]$$Separation of the cartesian components provides the desired relations.
 

1. What are polar coordinates?

Polar coordinates are a system for representing points in a plane using a distance from a fixed point and an angle from a fixed reference direction.

2. How are polar coordinates related to heliocentric ecliptic coordinates?

Polar coordinates and heliocentric ecliptic coordinates are both coordinate systems used to describe the position of objects in space. However, polar coordinates are based on a fixed point, while heliocentric ecliptic coordinates are based on the position of the sun.

3. What is the difference between polar coordinates and Cartesian coordinates?

While polar coordinates use a distance and angle to describe a point, Cartesian coordinates use a horizontal and vertical distance from a fixed reference point. This makes polar coordinates more suitable for describing circular or rotational motion.

4. Why are heliocentric ecliptic coordinates important in astronomy?

Heliocentric ecliptic coordinates are important in astronomy because they allow us to accurately track the movements and positions of objects in our solar system, such as planets, comets, and asteroids. They also help us understand the relationship between the Earth, sun, and other celestial bodies.

5. How do scientists convert between polar coordinates and heliocentric ecliptic coordinates?

To convert between polar coordinates and heliocentric ecliptic coordinates, scientists use mathematical equations that take into account the position of the sun, Earth, and other celestial bodies. This allows them to accurately translate between the two coordinate systems and plot the position of objects in space.

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