SME
Grade 12 Science · Problem type
A star’s mass sets its life
Use this for “A star has twice the Sun’s mass. How bright is it, and how long will it last?”
Diagram
The
star:
2
M☉
on
the
main
sequence
shines
L
=
2
3
×
√
2
=
11.3
L☉,
at
about
7,960
K.
It
burns
hydrogen
in
its
core
for
t
=
10
10
2
2
×
√
2
=
1.77 ×
10
9
years:
more
mass,
a
much
shorter
life.
This is an example. Type your own values to start a new problem.
t (years)
=
10¹⁰
years
÷
(
M (M☉)
)
2.5
L
Luminosity
worked out (not typed) · L☉
Formulas
L
=
M
3.5
11.3
=
2
3.5
t
=
10
10
M
2.5
1.77 ×
10
9
=
10
10
2
2.5
Assumptions
•
On the main sequence a star’s luminosity grows about as its mass to the power 3.5 (less steeply above about 20 M☉).
•
Its life there is its fuel (its mass) over the rate it burns it (its luminosity): the Sun’s 10¹⁰ years ÷ M2.5.
•
Masses and luminosities are in Suns.
•
Stars under about 0.8 M☉ outlive the universe so far.
Step-by-step
WE KNOW
M = 2 M☉
FIND
luminosity (L), main-sequence lifetime (t)
Step 1 · Find luminosity (L)
Use L = M3.5
Luminosity rises steeply with mass on the main sequence.
L
=
2
3.5
L
=
11.3
L☉
Step 2 · Find main-sequence lifetime (t)
Use t = 10¹⁰ ÷ M2.5
M
times
the
fuel,
burned
M
3.5
times
as
fast,
lasts
the
Sun’s
10
10
years
×
M
M
3.5
.
t
=
10
10
2
2.5
t
=
10
10
5.657
t
=
1.77 ×
10
9
years
Check
Put the numbers back in. Both sides should match.
11.3
≈
2
3.5
✓
1.77 ×
10
9
≈
10
10
2
2.5
✓
Topic check
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