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Topics: General relativity, Star, Derivative Pages: 5 (464 words) Published: December 28, 2013
Space Math Differentiation - "Ch..Ch..Ch..Changes"
There are many situations in
which differentiation has to be performed
on formulae in astrophysics. Many
objects such as stars and galaxies
display 'differential rotation' which leads
to many interesting and unusual
phenomena. The formula describing
these phenomena are usually 'differential
equations' that relate changes in one
quantity to changes in another.
Here are some popular equations
used
in
astrophysics
whose
differentiation will test your basic skills!
Image: Model of solar differential rotation
(Courtesy: Stanford Solar Center / NASA SOHO)

Problem 1 Find dm/dv - the rate of
change of mass with velocity near the
speed of light.
Problem 2 - Find dL/dT - The rate of
change of a star's luminosity with its
temperature.
Problem 3 - Find dR/dt - The rate of
change of the size of an expanding
supernova remnant with time (in other
words, its expansion speed!).
Problem 4 - Find dV/dz - The rate of
change of the apparent speed of a body
with its gravitational redshift.
Problem 5 - Find dD/dN - the rate of
change of the Debye shielding radius in a
plasma with a change in the density of the
plasma.
2

Problem 6 - Find (1/m ) d /dD - the rate
of change of the energy of empty space as
you change the number of dimensions to
space.

Space Math

http://spacemath.gsfc.nasa.gov

39

39

Answer Key
Problem 1 - Find dm/dv - the rate of change of mass with velocity near the speed of light. dm
------ =
dv

Mv
--------------------c2 (1- [v2/c2])3/2

Problem 2 - Find dL/dT - The rate of change of a star's luminosity with its temperature. dL
----- = 16
dT

R2

T3

Problem 3 - Find dR/dt - The rate of change of the size of an expanding supernova remnant with time (in other words, its expansion speed!).

dR
----- = 1/4 (3 mV/(
dt

))1/4 t -3/4

Problem 4 - Find dV/dz - The rate of change of the apparent speed of a body with its gravitational redshift. This requires using the quotient rule for differentiation 2
dU - (U/v ) dV

dV
---dz

1
= ---------------- 2(z+1)
(z + 1)2 + 1

-

1
------------------ 2(z+1)
[ (z+1)2 + 1]2

=

d(U/V) = (1/V)

2(z+1)3
-------------------[ (z+1)2 + 1]2

Problem 5 - Find dD/dN - the rate of change of the Debye shielding radius in a plasma with a change in the density of the plasma.
dD
---- = -1/2 (kT/4 e2)1/2 N-3/2
dN
Problem 6 - Find d /dD - the rate of change of the energy of empty space as you change the 5

3

2

number of dimensions to space. Again the quotient rule is needed where U = D - 3D + 2D 2
V= D + 8D + 16

1 d
--- ---- =
2
m dD

Space Math

4

2

5D - 9D + 4D
-----------------------2
8 (D + 8D + 16)

5

3

2

[D - 3D + 2D ] (2D + 8)
- -------------------------------2
2
[8 (D + 8D + 16)]

http://spacemath.gsfc.nasa.gov

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