a) A galaxy G is moving away radially with speed with respect to an observer O. The relation between fe, the frequency of light emitted at G, and fo, the frequency observed at O, is fo=fe 1-B 1 + ẞ' where = v/c (c is the speed of light). For ẞ <1 find a power series expansion of the above formula up to and including terms of order ẞ². b) Consider a spiral galaxy whose surface brightness is given by the law I(0) = IN(0) + ID(0), where IN(0) and ID (0) represent the surface brightnesses of the circular bulge and disc respectively. Their laws are IN (0) = 1(0)e-(0/0) 1/3 and ID(0) = I(0) 5 1000 ' where is the angular distance from the centre of the galaxy, I(0) is the central brightness of the bulge and 0, is the scale angle of the bulge. Calculate the numerical value of its total flux density in units of I(0)π02. c) Consider a galaxy with a density profile of the type ρο p(r): = [1+ (r/ro)3]2' where po is the central density and ro a reference radius. Find the circular velocity profile e(r) for such a galaxy.
a) A galaxy G is moving away radially with speed with respect to an observer O. The relation between fe, the frequency of light emitted at G, and fo, the frequency observed at O, is fo=fe 1-B 1 + ẞ' where = v/c (c is the speed of light). For ẞ <1 find a power series expansion of the above formula up to and including terms of order ẞ². b) Consider a spiral galaxy whose surface brightness is given by the law I(0) = IN(0) + ID(0), where IN(0) and ID (0) represent the surface brightnesses of the circular bulge and disc respectively. Their laws are IN (0) = 1(0)e-(0/0) 1/3 and ID(0) = I(0) 5 1000 ' where is the angular distance from the centre of the galaxy, I(0) is the central brightness of the bulge and 0, is the scale angle of the bulge. Calculate the numerical value of its total flux density in units of I(0)π02. c) Consider a galaxy with a density profile of the type ρο p(r): = [1+ (r/ro)3]2' where po is the central density and ro a reference radius. Find the circular velocity profile e(r) for such a galaxy.
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![a) A galaxy G is moving away radially with speed with respect to an observer O. The relation
between fe, the frequency of light emitted at G, and fo, the frequency observed at O, is
fo=fe
1-B
1 + ẞ'
where = v/c (c is the speed of light). For ẞ <1 find a power series expansion of the
above formula up to and including terms of order ẞ².
b) Consider a spiral galaxy whose surface brightness is given by the law
I(0) = IN(0) + ID(0),
where IN(0) and ID (0) represent the surface brightnesses of the circular bulge and disc
respectively. Their laws are
IN (0) = 1(0)e-(0/0) 1/3 and ID(0) =
I(0)
5
1000
'
where is the angular distance from the centre of the galaxy, I(0) is the central brightness
of the bulge and 0, is the scale angle of the bulge. Calculate the numerical value of its total
flux density in units of I(0)π02.
c)
Consider a galaxy with a density profile of the type
ρο
p(r):
=
[1+ (r/ro)3]2'
where po is the central density and ro a reference radius. Find the circular velocity profile
e(r) for such a galaxy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb360a251-a88e-4e24-b6e1-ceeebc35a45e%2F25c38936-90ca-462e-83ae-2cbe05125628%2F33wqzj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) A galaxy G is moving away radially with speed with respect to an observer O. The relation
between fe, the frequency of light emitted at G, and fo, the frequency observed at O, is
fo=fe
1-B
1 + ẞ'
where = v/c (c is the speed of light). For ẞ <1 find a power series expansion of the
above formula up to and including terms of order ẞ².
b) Consider a spiral galaxy whose surface brightness is given by the law
I(0) = IN(0) + ID(0),
where IN(0) and ID (0) represent the surface brightnesses of the circular bulge and disc
respectively. Their laws are
IN (0) = 1(0)e-(0/0) 1/3 and ID(0) =
I(0)
5
1000
'
where is the angular distance from the centre of the galaxy, I(0) is the central brightness
of the bulge and 0, is the scale angle of the bulge. Calculate the numerical value of its total
flux density in units of I(0)π02.
c)
Consider a galaxy with a density profile of the type
ρο
p(r):
=
[1+ (r/ro)3]2'
where po is the central density and ro a reference radius. Find the circular velocity profile
e(r) for such a galaxy.
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