Solving Ellipses on Spheres: Finding Point Sources in Galaxies

• Sleeker
In summary, to determine whether a point source is within an ellipse-shaped galaxy, convert the coordinates of the galaxy's center and the point source into Cartesian coordinates and calculate the distance between them. Then, compare this distance to the major and minor axes of the ellipse. If the distance is less than the minor axis or greater than the major axis, the point source is within the ellipse. If it is between the two, use the eccentricity of the ellipse to determine whether it is within or outside.
Sleeker
I have galaxies in the shapes of ellipses, and I have point sources around these galaxies. I need to find a formula to determine whether or not the point sources are within a specific galaxy or outside. It would be extremely simple with circles, but I can't figure it out with ellipses.

Given:
Major and minor axes of the ellipse in arcminutes.
Angle of major axis relative to north (measured to the east).
Center of galaxy (measured in RA and DEC, easily convertible to azimuth and inclination).
Location of point sources (measured in RA and DEC, easily convertible to azimuth and inclination).

My attempt:
If I set the center of the galaxy and the location of the point source as two vectors (radius = 1, RA = azimuth, 90-DEC = inclination), I can use the dot product to obtain the angle between the galaxy's center and the point source. I just cannot seem to figure out how to check whether or not the point source is within the ellipse or not. It's trivially easy if the distance is less than the minor axis or greater than the major axis, but very difficult if it lies between the two.

The difficult parts are trying to map the ellipse onto the inner surface of the sphere and trying to determine the direction from the galaxy's center to the point source (and then how to use that direction to help me), that is, if I even need to do these things.

My first attempt basically ignored the non-Euclidean nature of the surface and failed at high DECs.

(RA is like longitude and DEC is like latitude.)

A better approach would be to convert the RA and DEC coordinates of both the galaxy's center and the point source into Cartesian coordinates (x,y,z). Then, calculate the distance between the two points (d) using the Pythagorean theorem. Finally, compare d to the major and minor axes of the ellipse. If d is less than the minor axis, or greater than the major axis, then the point source is within the ellipse. If it is between the two, then you can use the eccentricity of the ellipse to determine whether or not it is within the ellipse. The eccentricity is equal to the ratio of the major axis to the minor axis. If d is lower than the eccentricity multiplied by the minor axis, then the point source is within the ellipse. Otherwise, it is outside.

1. What is the purpose of solving ellipses on spheres?

The purpose of solving ellipses on spheres is to identify and locate point sources in galaxies. These point sources can be stars, quasars, or other celestial objects that emit light. By accurately determining the location of these point sources, scientists can gain a better understanding of the structure and dynamics of galaxies.

2. How are ellipses and spheres related in this context?

In this context, ellipses and spheres are related because the projection of a spherical galaxy onto a 2-dimensional image results in elliptical shapes. This is due to the combination of the galaxy's rotation and its orientation in the sky. By solving for the ellipses on the sphere, scientists can accurately map the 3-dimensional structure of the galaxy.

3. What techniques are used to solve ellipses on spheres?

The most common technique used to solve ellipses on spheres is through a process called "ellipse fitting." This involves fitting an ellipse to the observed data points and determining the parameters that best describe the ellipse's shape and orientation. Other techniques, such as Markov chain Monte Carlo methods, can also be used to solve for ellipses on spheres.

4. What challenges do scientists face when solving ellipses on spheres?

One of the main challenges when solving ellipses on spheres is the presence of noise in the observed data. This can be caused by factors such as atmospheric turbulence, telescope limitations, or instrumental effects. Another challenge is the potential for degeneracies, where multiple ellipses can fit the same data points, making it difficult to determine the true shape and orientation of the galaxy.

5. How does solving ellipses on spheres contribute to our understanding of galaxies?

By accurately determining the locations of point sources in galaxies, scientists can create detailed maps of the galactic structure. This can provide insights into the distribution of stars, gas, and dark matter within the galaxy. Additionally, studying the rotation and orientation of galaxies can help us understand their formation and evolution over time.

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