1. Is Dark Matter the Only Possible Solution? Ohio University - Lancaster Campus slide 1 of 13 Spring 2009 PSC 100

2. Ohio University - Lancaster Campus slide 2 of 13 Spring 2009 PSC 100 In 1983, Mordehai Milgrom, a professor at the Weismann Institute in Israel, proposed a startling alternative to dark matter. It was a revision of Newton’s 2 nd Law of Motion. Milgrom called it MOND: Modified Newtonian Dynamics www.weizmann.ac.il

3. Ohio University - Lancaster Campus slide 3 of 13 Spring 2009 PSC 100 Newton’s Law of Gravity states that the force between two objects comes from the product of their masses, and is proportional to the inverse of the square of the distance between them… Force = Gm 1 m 2 r 2 G is a proportionality constant, 6.67 x 10 -11 m 3 /kg▪s 2 The fact that G is so small indicates that gravity is a weak force.

4. Ohio University - Lancaster Campus slide 4 of 13 Spring 2009 PSC 100 Newton’s 2 nd Law also states that a force on an object is given by the object’s mass and the acceleration applied to it… F = m▪a

5. Ohio University - Lancaster Campus slide 5 of 13 Spring 2009 PSC 100 What Milgrom proposed was that when the distance between two objects was very large, then the acceleration would be very small, but not as small as Newton predicted. Newton’s 2 nd law changes to… F = m▪a 2 a 0 where a 0 is 1.2 x 10 -10 m/s 2

6. Ohio University - Lancaster Campus slide 6 of 13 Spring 2009 PSC 100 a 0 is a constant derived from the speed of light and the age of the universe. If an object had started at rest and accelerated with an acceleration a 0 since the beginning of the universe, today it would be moving at c, the speed of light, 3.00 x 10 8 m/s. More importantly… then F = Gm 1 m 2 = ma 2 r 2 a 0

7. Ohio University - Lancaster Campus slide 7 of 13 Spring 2009 PSC 100 Since, over great distances, the gravitational force is proportional to a 2, not just a, less mass is needed to account for the force. In other words, Milgrom’s equation eliminates the need for dark matter.

8. Ohio University - Lancaster Campus slide 8 of 13 Spring 2009 PSC 100 How does Milgrom’s equation eliminate the need for dark matter in our galaxy? Diagram Credit: Phil Hibbs, Michael Gmirkin

9. Ohio University - Lancaster Campus slide 9 of 13 Spring 2009 PSC 100 Gm 1 m 2 = ma 2 r 2 a 0 Let’s substitute in m galaxy for m 1 and m sun for m 2 and the m on the right, then we can cancel out the m sun on both sides: GM = a 2 r 2 a 0

10. Ohio University - Lancaster Campus slide 10 of 13 Spring 2009 PSC 100 GMa 0 = a 2 Now take the square r 2 root of both sides: GMa 0 = a r

11. Ohio University - Lancaster Campus slide 11 of 13 Spring 2009 PSC 100 Finally, set this equation equal to the equation that relates velocity to acceleration for a star in a circular orbit around the galaxy’s center: a = GMa 0 = v 2 r r and cancel out the r’s on both sides: GMa 0 = v 2

12. Ohio University - Lancaster Campus slide 12 of 13 Spring 2009 PSC 100 Take the square root of both sides: GMa 0 = v What we understand from this, is that the velocity of a star orbiting the center of the galaxy has no dependence on its distance, r, from the center of the galaxy. And NO dark matter was needed. 4

13. Ohio University - Lancaster Campus slide 13 of 13 Spring 2009 PSC 100 Does this mean that the idea of dark matter is worthless? Many astrophysicists agree that Milgrom’s equation predicts and describes the rotational behavior of the galaxy (and many other low acceleration situations also!) But most astrophysicists don’t agree that the equation explains the cause of the behavior. Most astrophysicists remain convinced that the equation just coincidentally coincides with the behavior of dark matter. What do you think?