Flux, Luminosity and Inverse Square Law

Luminosity and Flux Here we review the definitions of some important quantities that we will use often in the course. Luminosity, L, is the total energy radiated from an object per second, measured in Watts. Energy flux is the flow of energy out of a surface, measured in Watts/m2. The observed flux (apparent brightness) of an object is the power we receive from it, depending on the distance to the object, measured in W/m2. We often just use the term flux without the "energy" or "observed" qualifier. The unspecified qualifier is determined by the context. Inverse Square Law How is the Observed Flux determined? Make a sphere of radius, d, around an object which is radiating power (such as the sun or a light bulb). All energy radiated from the object does not depend on the size of the sphere. However, the flow of energy per m2 passing through the sphere decreases as the size of the sphere increases. where, R = radius of the object L = luminosity of the object T = surface temperature of the object σ = Stefan-Boltzmann constant f1 = flux at distance d1 f2 = flux at distance d2 Inverse Square Law tells us how the energy flux changes depending on the distance from the object.

Magnitude and Distance Modulus
Apparent Magnitude The apparent magnitude gives how bright an astronomical object appears to an observer on Earth regardless of its intrinsic brightness. Letf be the observed flux from object. Then the apparent magnitude difference between two objects 1 and 2 is given by, Note, the brighter object has a smaller magnitude!	Absolute Magnitude It is defined as the apparent magnitude of an object placed at a distance of 10 parsecs (pc). By the relation between flux and luminosity, For an identical object, L1=L2=constant. Let's d be the actual distance to the object, and d2=10pc if we set m2 as an absolute magnitude of the object. Now, gives us,
Distance Modulus The logarithm of the ratios of apparent to absolute brightnesses, equal to the difference of absolute magnitude to apparent magnitude (see the final expression in absolute magnitude part). Distance Modulus, μ is defined by,

Hertzsprung-Russell (HR) Diagram
In about 1910, having investigated the effect of the temperature of an object and the color of radiation given off, several scientists reasoned that it should also be possible to relate the temperature of a star to its luminosity. If all stars were alike, stars of the same temperature would give off the same amount of light. Hotter stars would be brighter than cooler stars. The first diagram (luminosities (absolute mag.) vs. surface temperatures (color) of stars) was plotted by Ejnar Hertzsprung in 1911, and (independently) by Henry Norris Russell in1913. This diagram came to be known as the Hertzsprung Russell diagram, or simply the HR diagram.
	[Figure] Hertzsprung-Russell Diagram for stars in Solar neighborhood The horizontal scale shows the temperature of the stars. It is by convention reversed so that the hottest stars are located near the origin, and the coolest stars are to the right. It is a "two-dimensional" plot for the observed stars, and represents one of the greatest observational syntheses in astronomy and astrophysics! Measuring the stars luminosity was more difficult. Some stars are closer to the earth, while others are extremely distant. Because light becomes more spread out as it moves away from its source, more distant stars would look dimmer, even if they were very bright up close. It was decided to standardize the brightness of each star so that it appeared to be located at a distance of 10 parsecs from the earth, or about 32.6 light years away. The brighntess of the sun would be set at "1", and other stars would be ranked accordingly.
It is readily apparent that the H-R Diagram is not uniformly populated, but that stars preferentially fall into certain regions of the diagram. The majority of stars fall along a curving diagonal line called the main sequence but there are other regions where many stars also fall. The HR Diagram may be partially understood in terms of the luminosity for an object emitting thermal radiation (L ~ R2T4). If all objects in the HR Diagram were the same size then all objects would lie along a diagonal line in this plot. There are different regions main sequence, giant, supergiant, etc. Most stars lie along the main-sequence. For a given spectral class (e.g. K), there can be more than one luminosity, i.e. main-sequence, giant or supergiant. On the main sequence, there are many more K and M stars than O and B stars.
	Luminosity Class Ia : Brightest Supergiants Ib : Less luminous supergiants II : Bright giants III : Giants IV : Subgiants V : Main-sequence stars Total energy per second radiated from a star of radius R. The luminosity, L, is given by Stefan-Boltzmann equation; So supergiants must be big!!!

Application - Distances to Stellar Clusters
	MAIN SEQUENCE FITTING Make an HR diagram with Apparent Magnitude and Spectral Class of a cluster (open or globular) of stars. Overlay with the standard HR diagram to compare the Apparent and Absolute Magnitudes of the MS. This gives m - M which can be used to find the cluster distance. m - M = 5 log ( d / 10 )