Recently, a video clip of a confrontation between players from the Orlando Magic basketball team went viral. The incident occurred during a game against the Philadelphia 76ers, and it showed tension escalating between two teammates, Markelle Fultz and Michael Carter-Williams. In the clip, Fultz can be seen yelling animatedly at Carter-Williams, while the latter responds with aggression. The heated exchange was quickly broken up by their teammates and coaches. The main idea of this clip is the confrontation between Fultz and Carter-Williams. It is interesting to note that Fultz and Carter-Williams are both point guards for the Orlando Magic, and they were caught in a heated argument during a game.
In casting the circle, the magician moves deosil, or clockwise, to the motion of the Sun, the Moon, and the stars through the sky. For black magic rituals, the magician moves widdershins, or counterclockwise. He leaves a small opening and then steps inside, carefully closing the opening to prevent clever demons from slipping inside. The magician consecrates the circle with the four Elements and invokes the guardian spirits who watch over the four quarters of the sky (the cardinal points) and the four elements. (See Lords of the Watchtower.)
The magician consecrates the circle with the four Elements and invokes the guardian spirits who watch over the four quarters of the sky the cardinal points and the four elements. The magician consecrates the circle with the four elements, Earth, Air, Water and Fire, and invokes the guardian spirits who watch over the four quarters of the sky the cardinal points and the four elements.
It is interesting to note that Fultz and Carter-Williams are both point guards for the Orlando Magic, and they were caught in a heated argument during a game. The incident, caught on camera, sparked a lot of speculation and discussion among fans and analysts. There are various theories being discussed about the cause of the confrontation.
Measuring Stellar Sizes
Most stars fall along a straight line in the HR diagram, running from upper-left to lower-right. That tells us that most stars must fall within a relatively narrow range of sizes.
But is there any way we can measure the size of a star more directly? After all, real stellar spectra aren't exactly like those of blackbodies, so these rough estimates might be wrong.
- Direct imaging
- Lunar occultation
- Eclipsing binaries
- Interferometry
As we discuss each of the methods below, keep in your mind a single, concrete example: could this method determine the size of an ordinary star like the Sun, if it happened to be a mere 10 parsecs away from us?
Q: What is the angular size of a sun-like star at a distance of 10 pc? Express your answer in radians and in milli-arcseconds (mas).
Direct imaging
It seems obvious: if you want to measure the size of a star, just point your telescope at it and take a picture. Measure the angular size of the star in the image, then multiply by the distance to find the true linear diameter. What's so hard about that?
The problem is a phenomenon called diffraction. If you use a real optical system with an aperture of diameter D, light rays won't come to a perfect point at the focus; instead, interference from rays entering the aperture at different locations will form a fuzzy blob of light, surrounded by a series of faint rings. The image below was taken by a telescope with a mirror made of liquid mercury, 1.5 meters in diameter.
This pattern is called the "Airy pattern" after George Airy, Astronomer Royal of England, who first derived the angular size of the central blob and the rings which surround it. The intensity and spacing of the rings are related to the Bessell function of first order, J1. For our purposes, the important item is the angular size of the central blob, from its center to the first minimum in the diffraction pattern.
Q: In order to resolve a sun-like star at 10 pc, using visible light (5500 Angstroms), how large would a telescope's aperture have to be?
It helps to work as far into the ultraviolet as possible, since the decrease in wavelength λ shrinks the Airy pattern. Unfortunately, the Earth's atmosphere prevents ultraviolet light from reaching the ground. Astronomers have used telescopes in space to take near-UV images of a very few stars in hopes of resolving them. In at least one case, they have succeeded -- barely. Here's an image from the press release:
And here are a pair of radial profiles, of Betelgeuse and an unresolved star, from a conference proceeding describing the observations.
Q: The parallax of Betelgeuse is 0.00764 arcsec. What is the diameter of Betelgeuse? Express your answers in terms of the solar diameter.
Lunar Occultations
- find a star which will be covered by the Moon as it moves through the sky
- using a high-speed device, measure the light from the star as a function of time
- calculate the size of the star from the light curve
Sounds easy, right? Just watch as the moon's limb covers the star:
The time it takes for the star's light to disappear, Δt,
is just the angular size of the star θ divided by the angular speed v of the Moon through the sky.
Q: What is the Moon's angular speed through the sky? Express your answer in arcseconds per second. Q: How long would it take for the Moon's limb to cover a star of angular diameter 100 mas?
- a high-speed photometer or camera, capable of hundreds of measurements per second .
- and a big enough telescope to gather enough photons within each frame to make a decent measurement
This requirement of collecting lots of photons in a very short time is a killer. The lunar occultation method is therefore restricted to relatively bright stars. It's also restricted to stars which happen to lie near the ecliptic, of course.
But . it's even worse! It turns out that diffraction makes life difficult for astronomers again. As the Moon's limb begins to pass in front of the star's disk, it diffracts the light from the star. As an analogy, you know the diffraction pattern of light passing through a narrow slit.
As the Moon's limb approaches and covers a star, we on the Earth see something like this pattern of alternating dark and light spots moving past our detectors. So the light curves (intensity versus time) that we record will be somewhat more complicated than the simple decreasing curve shown earlier.
- unresolved (point-like) stars will show a pure diffraction pattern of fringes
- resolved (disk-like) stars will show a smoothly decreasing curve
- partially-resolved stars will show a combination of the two extremes
The mathematical analysis of these real light curves is a complicated business. Once again, Bessel functions come into play. One must also make some assumptions about the star itself: is it a uniformly bright disk with a sharp edge, or is it limb-darkened like the Sun?
The bottom line is that the lunar occultation method is effective at measuring the angular diameters of stars down to about 1 mas. Back in 1987, White and Feierman published a catalog angular diameters of 124 stars measured by lunar occultations. A paper by Richichi states that another 50 stars were measured in this way up to 1998. The lunar occultation method has very clear limits, but clearly has given us a great deal of information on stellar sizes.
Eclipsing binary stars
Fortunately, the Moon isn't the only "moving knife edge" we can use to determine the sizes of stars. There are many instances in which we can use one star as the "moving knife edge" to measure the size of a second star. All we have to do is find an eclipsing binary system: a pair of stars orbiting around each other, oriented in space so that one star passes in front of the other as seen from the Earth. If we measure the light coming from such a system carefully, we can detect the decrease in total intensity as a portion of one of the stars is covered:
- what if the plane of the orbit isn't exactly edge-on as seen from the Earth?
- what if the two stars aren't the same size?
Wait a minute -- we need to know the orbital speeds of the two stars, VA and VB. How can we determine that?
Yes -- use spectra of the system to measure the Doppler shift of each star's absorption lines. If the orbit is circular, then the maximum radial velocity we measure for each star is exactly the same as the tangential velocities VA and VB with which they pass in front of each other.
The star CV Velorum, or CV Vel for short, is an eclipsing binary which shows features from both stars in its spectra. The period of its light curve is P = 6.889 days. Here's a sample of its light curve, from the paper Four-colour photometry of eclipsing binaries. VIII - CV Velorum, light curves, photometric elements and absolute dimensions
Here are a few small segments of its spectrum, taken at a time ("quadrature") when the components were moving nearly directly towards us and away from us, from the paper Spectroscopic observations of eclipsing binaries. III - Definitive orbits and effects of line blending in CV Velorum In this system, the two stars have nearly the same mass, so they orbit their center of mass with nearly the same velocity. The features you see in the spectrum below are shifted by equal amounts away from the rest wavelength of each line, which is halfway between the features.
Q: What are the orbital speeds of the primary and secondary stars? Q: What is the duration of the primary eclipse? Q: Assume that the two stars are the same size. What is the diameter of each one?
Interferometry
A topic for another day.
For more information
So, just how many stars have well-measured diameters? One estimate comes from the Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition -Comments and statistics. You can query this catalogue using the Vizier facility. The catalog contains diameters for over 9700 stars. Is it a fair sample of stellar sizes? Take a look at the distribution of measured sizes, in terms of the solar diameter:
What fraction of these stars are smaller than the Sun? What fraction of all stars in the galaxy are smaller than the Sun? There is clearly a bias towards large stars -- why?
Copyright © Michael Richmond. This work is licensed under a Creative Commons License.
Some believe it may have been due to frustration over a play or miscommunication on the court. Others speculate that there may have been tension brewing between the two players for some time. Whatever the cause, it is clear that emotions were running high and tempers flared in the heat of the moment. This incident highlights the challenges that professional athletes face, both physically and mentally. The pressure to perform at a high level, combined with the competitive nature of sports, can sometimes lead to confrontations and disagreements among teammates. However, it is important for teams to address these issues quickly and find ways to resolve conflicts in a healthy and constructive manner. The Orlando Magic organization responded to the incident by stating that they would handle the matter internally and work towards finding a resolution. It remains to be seen how the team will address the issue and whether any disciplinary actions will be taken. Overall, the Orlando Magic confrontation clip has brought attention to the dynamics of teamwork and the challenges that come with competing at a professional level. It serves as a reminder of the importance of communication and conflict resolution in sports and underscores the need for teams to address issues promptly to maintain a positive and cohesive environment..
Reviews for "Orlando Magic Confrontation Clip: Shining a Light on Leadership Issues"
1. John - 1/5
I was extremely disappointed by the "Orlando Magic confrontation clip". I found it to be quite tasteless, as it seemed to glorify violence and aggression. The content of the clip felt forced and contrived, lacking any real substance or depth. It did nothing to enhance my understanding or appreciation of the Orlando Magic or basketball as a whole. Overall, I cannot recommend this clip to anyone who values meaningful and thought-provoking content.
2. Sarah - 2/5
I have to say I was not impressed with the "Orlando Magic confrontation clip". While the concept of showcasing an intense moment during a basketball game was intriguing, the execution fell flat. The clip lacked cohesion and failed to engage the viewer. Additionally, the use of excessive slow-motion shots and dramatic music felt forced and overdone. I expected more from a clip focusing on a professional sports team, and unfortunately, this one missed the mark for me.
3. David - 1/5
The "Orlando Magic confrontation clip" was nothing more than a glorified display of aggression and bad sportsmanship. I found it disturbing how the clip seemed to promote violence and conflict, rather than highlighting the skill and talent of the players. It left a sour taste in my mouth and did nothing to change my perception of the Orlando Magic in a positive way. I would not recommend this clip to anyone looking for a genuine appreciation of the sport or the team.
4. Emily - 2/5
The "Orlando Magic confrontation clip" felt like a missed opportunity to showcase the team's remarkable abilities and unity. Instead, it focused solely on a negative, confrontational moment that did not represent the team's overall spirit or values. I was hoping for a more balanced and insightful portrayal of the Orlando Magic, but unfortunately, this clip fell short of my expectations. While it may appeal to those who enjoy controversial and intense sports moments, I believe it did a disservice to the team as a whole.