- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 1 (121.37 KiB) Viewed 9 times
- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 2 (214.45 KiB) Viewed 9 times
- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 3 (230.88 KiB) Viewed 9 times
- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 4 (58.23 KiB) Viewed 9 times
- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 5 (144.61 KiB) Viewed 9 times
- Case 1 1 Two Stars Of Equal Surface Brightness Same Brightness Per Unit Area 2 No Limb Darkening 3 Partial Ecli 6 (87.09 KiB) Viewed 9 times
CASE 1: (1) two stars of equal surface brightness (same brightness per unit area) (2) no limb darkening (3) partial eclipses (4) orbit is either circular, or elliptical with the major axis (line connecting the foci) along the line of sight of the observer. LIGHT CURVE 1 LIGHT INTENSITY A B с D Close-up view of eclipsing system. E A G H time between minima one period J The arrows point toward the Earth. L M TIME E View from Earth. (front view) C View from above the orbital plane of the binary system could be either one of these two possible cases. Q Note how the major axis (the line along the longest length of the ellipse) lies along the observer's line of sight. View from above. (top view) Each portion of the light curve may be explained in detail as follows: (A) Light is at maximum intensity as the two stars are side by side. (B) Light intensity decreases gradually (sloping line) as one star begins to move in front of the other. (C) Light intensity reaches its minimum point. as the most of one star that will ever be blöcked off, is blocked off. This is always true for partial eclipses (condition 3). (D) Light intensity increases gradually as one star moves out from in front of the other. (E) Same as A. (F) Same as B, except the star which originally was lhe did the covering (C) Samo o but
Each portion of the light curve may be explained in detail as follows: (A) Light is at maximum intensity as the two stars are side by side. (B) Light intensity decreases gradually (sloping line) as one star begins to move in front of the other. (C) Light intensity reaches its minimum point. as the most of one star that will ever be blöcked off, is blocked off. This is always true for partial eclipses (condition 3). (D) Light intensity increases gradually as one star moves out from in front of the other. (E) Same as A. (F) Same as B, except the star which originally was covered is now passing in front of the star which originally did the covering. (G) Same as C, but with roles reversed. (H) Same as D, but with roles again reversed. (1) Same as A. (J) Same as B. (K) Same as C, etc. Note that the TIME C to K is the time of one complete cycle or period. This could be measured in anything from hours to years, depending on the system observed. This is from the time one star is in front of the other until the next time that same star is in front of the other again. Note also that the time from C to G is the same as the time from G to K, indicating equal times between eclipses. (Always true for condition 4 above.) And finally, note that minimum G has the same depth as minimum C. (Always true for stars of equal brightness: condition 1). For convenience we assume one star to be at rest and the other to be in orbit around it. This is a perfectly legitimate reference frame. When considering motions, we often picture each star in orbit about the barycenter, however. CASE 2: (1) stars of equal surface brightness (2) no limb darkening (3) partial eclipses (4) major axis not lined up with observer's line of sight. Only condition (4) has changed from CASE 1. LIGHT CURVE 2 LIGHT INTENSITY A B C V₁T H G E D F shorter time 2 1 longer time one period J K M View from above the orbital plane of the stars. O O O Q G K TIME The arrows point toward Earth. Top view. Note that the only change is that the time from C to G is less than the time from G to K. The reason for this is related to varying orbital speeds as described by Kepler's laws. As one star moves from C to G, as shown below the light curve, it has a relatively short distance to go and it is moving quite rapidly. Just the reverse is true as it moves from G to K. When you draw a light curve for such a case, make the short interval VERY short, and the long two or three times as long, so there is no question you understand the correct characteristic of the pattern. Note also that AT LEAST THREE, and preferably four minima must be drawn for a complete curve. SIDES SHOULD ALWAYS BE ANGLED, NEVER VERTICAL, i.e. make them into a V shape. Note: The minimum is the low point of the graph. (Plural: minima) 3
CASE 3: (1) stars of equal surface brightness (2) no limb darkening (3) total and annular eclipses (4) circular orbit. This time condition (3) was changed from CASE 1. We will assume we see the orbital plane from exactly edge on. Since the two stars will almost certainly be somewhat different in size, we expect to see alternately annular and total eclipses. The effect on the light curve is the same in either case. LIGHT CURVE 3 LIGHT INTENSITY A LIGHT INTENSITY B с LIGHT CURVE 5 LIGHT INTENSITY E D F G one period H 1 TIME Close up of physical situation as seen from the DIRECTION of Earth. TOTAL ECLIPSE: Large star in front of small star. K Note that we have flat bottomed minima rather than pointed ones. This indicates alternately annular and total eclipses. These always go together, i.e. if one eclipse is annular the other is total, and both are flat bottomed. And if one is partial so is the other, and both have points. The length of the flat bottom will depend on how long one star is completely blocking light from the other. These lengths are the same for both the total and annular portions of the cycle for the same pair of stars. M CASE 4: Draw a light curve for the following conditions: (fourth case): (1) two stars of equal surface brightness (2) no limb darkening (3) alternately total and annular eclipses (4) elliptical orbit with a major axis not lined up with the observer's line of sight. DRAW IN AT LEAST FOUR MINIMA. Exaggerate all important characteristics. The sides of the minima MUST slope in. They should come down, and go up, at an angle, left to right. Do your practice sketch here. Draw your final work on page 11 using a ruler and measuring carefully. LIGHT CURVE 4 4 ANNULAR ECLIPSE: Small star in front of larger one. Front view. CASE 5: Now we will vary condition 1. Consider two stars which differ in surface brightness (brightness per unit area). Suppose we have a small star of high surface brightness, and a large star of low surface brightness, e.g. a white dwarf and a red giant. When the small star is in front of the large star, each square inch of the large star that is blocked is replaced by a square inch of the small star that sends us more light than the covered area of the big star alone, but less than that sent by both stars together. When the small star is eclipsed by the large star, the minima will be deeper than the other minima because the relatively bright area of the small star is covered by a relatively dim area of the large star. The total time from start to end of each eclipse is the same in both cases, however. Conditions for the fifth case: (1) Stars with different surface brightness (2) no limb darkening (3) alternately total and annular eclipses (4) aligned elliptical orbit. √ TIME
CASE 6: Draw a light curve for the following conditions (sixth case): (1) a small star of high surface brightness and a large star of low surface brightness (2) no limb darkening (3) partial eclipses (Sides of graph MUST slope in.) (4) circular orbits. LIGHT CURVE 6 LIGHT INTENSITY TIME CASE 7: Draw a light curve for the following conditions: (1) two stars of equal surface brightness (2) both stars have limb darkening (3) alternately total and annular eclipses (Sides of graph MUST slope in.) (4) orbit is elliptical with major axis not aligned with the observer's line of sight. LIGHT CURVE 7 LIGHT INTENSITY 11 TIME
When we studied the Sun we learned that the term limb darkening means the Sun is darker around the edges than at the center. The same effect is encountered in stars other than our Sun. Thus less light is lost from covering the edges of the star than from covering the middle. The result is a gradual curve into the eclipse rather than sharp corners. The shape of the curve would change as shown. Partial eclipse without limb darkening. Total or annular without limb darkening. LIGHT CURVE 7 LIGHT INTENSITY CASE 7: Draw a light curve for the following conditions: (1) two stars of equal surface brightness (2) both stars have limb darkening (3) alternately total and annular eclipses (Sides of graph MUST slope in.) (4) orbit is elliptical with major axis not aligned with the observer's line of sight. Do your practice sketch here. Draw your final work on page 11 using a ruler and measuring carefully. Partial eclipse with limb darkening. LIGHT CURVE 8 Total or annular with limb darkening. LIGHT INTENSITY All corners are curved, but the lower corners curve into a FLAT bottom. The sides slope in and are NOT vertical. Note that the top corners are curved, but the minimum is still a point. 6 Sometimes a dimmer star reflects radiation from a brighter star just before going into or just after coming our of an eclipse. Under these conditions the light curve will rise above its normal maximum. CASE 8: (1) two stars of surface brightness so different that reflection effects are important (2) no limb darkening (3) partial eclipses (4) circular orbits Fran TIME reflection effects
Light Curve of ß Persei (Algol) 2. 3. 4. -10 5. first minimum (0 hours) 0 10 EXPLAIN 20 second minimum (34.5 hours) 30 40 TIME (hours) Is there limb darkening? EXPLAIN Are any reflection effects obvious? EXPLAIN third minimum (69 hours) 50 60 EXPLAIN EACH ANSWER IN TERMS OF THE SHAPE OF THE GRAPH. 1. Does the light curve above picture: (A) a circular or an aligned elliptical orbit OR (B) an unaligned elliptical orbit? EXPLAIN How do the surface brightnesses of the two compare? EXPLAIN 2.0 10 2.3 A 2.6 I MAGNITUDE 2.9 D 3.2 Remember that total and annular eclipses always go together, and partials are always found with partials. Are the eclipses total and annular, or partial? 3.5 70