Lesson6v2_april

Lesson6v2_april - Astro120 QR Class 6(Mar/Apr-08 Orbital...

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Astro120 – QR Class 6 (Mar/Apr-08) Orbital velocity and escape velocity The orbital velocity v at a distance r from the center of a body of mass M is: r GM v = The escape velocity from the surface of a body of mass M r 2GM v = ( Note the factor of 2 in the square root!) A black hole containing a mass M has an escape velocity of the speed of light c: Sch R 2GM c = R is the radius of the black hole (called the Schwarzschild radius) Sch This can also be written as 2 Sch c GM 2 R = Special Relativity : Time dilation (moving clocks run slow) Length contraction (moving objects appear shorter) 2 2 O c v - 1 T T = 2 2 O c v 1 L L × = Note: 2 2 c v 1 is ALWAYS less than one 1
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Worked Examples 1. Superluminal Motion We will try to show how jets from AGN can have apparent superluminal motion (appear to move faster than c). Consider the diagram below. In 2004 we see the light that was emitted from a blob of material (ejected from a quasar) at position A and in 2007 we see the light emitted when the blob was at position B. If we measure the (apparent) distance from A to B to be 4 light years, then (a) how fast does the blob appear to move ? (b) If A is 3 light years further away from us than B, then what is the actual distance from A to B ? (c) How fast is the blob actually traveling at ? v A B 4 Lyr 3 Lyr 2004 2007 C EARTH 2
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(a) How fast does the blob appear to move ? Apparent distance from A to B = 4 light years time interval between seeing blobs = 2007-2004 = 3 years apparent speed = c 3 4 years 3 1year c 4 years 3 Lyrs 4 = × × = where c = speed of light (b) What is the actual distance traveled by the blob from A to B ? Using Pythagoras Theorem for the right angle triangle ABC 2 2 2 2 lyr) ( 25 lyr) 4 ( lyr) 3 ( AB) ( = + = Gives: AB = 5 light years (c) How fast does the blob actually move ? To find the actual velocity of the blob we need to know the time it takes
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This note was uploaded on 11/04/2008 for the course ASTRO 1002 taught by Professor Oswalt during the Spring '08 term at FIT.

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Lesson6v2_april - Astro120 QR Class 6(Mar/Apr-08 Orbital...

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