## Understanding stellar evolution

**The goals**

Students recognize the principal stellar observables (brightness and
color) and the relationships between them (Hertzsprung Russell diagram).
After a short introduction to stellar evolution theory, students verify
the Main Sequence Mass Luminosity Relationship
($L \propto M^{3\text{.}5}$) and calculate Main Sequence Lifetimes.

**The main items**

At the beginning of the twentieth century two astronomers, the Danish E.
Hertzsprung and the American H. N. Russell, established a correlation
between two important stellar parameters: brightness and color.

Since ancient times, the brightness of a star is indicated by
\"magnitudes\": I, II and so on, dim stars having larger magnitudes. We
don't confuse the relative magnitude with absolute magnitude. The
*relative magnitude* measure the brightness of a star as so as it
appears in the sky and it depends on the brightness and on the distance;
if we put a star at the distance of 10 Parsec (33 year light), its
magnitude is called *absolute magnitude*: obviously it depends only on
brightness of the star.

According to color, stars are grouped in classes: O, B, A, F, G, K, and
M. O stars are bleu and M are red, see table 1. Classes are oddly
sequenced because they were assigned long before we understood their
relationship to surface temperature: O stars are the hottest and M the
coolest ones; O and B stars are rare but very bright; M stars are
numerous but dim.

Hertzsprung and Russell discovered that in a Color-Magnitude diagram
(H-R diagram) stars appear distributed in well defined groups, so that
there is a correlation between temperature and luminosity (Fig. 1).

We think that it is important the student appreciates the difference
between a random distribution of two variables and a correlated one.

![](media/image1.png){width="5.0in" height="3.761023622047244in"}

![](media/image2.png){width="5.2in" height="3.97007874015748in"}

The H-R diagram provides a full interpretation in terms of stellar
evolution theory. With reference to fig. 1, a short of stellar evolution
says:

1.  The star is just born: its temperature rises, but not its
    luminosity. During this phase stars the thermonuclear reactions
    involving the burning of Hydrogen.

2.  The star is into main sequence: this is a period of stability in
    which the star burns the Hydrogen producing Helium. The star spend
    mostly time in the main sequence; we can calculate how long does the
    star stay into main sequence, using this easy equation:

$$t_{m\text{.}s\text{.}} = \text{10}^{\text{10}} \cdot \left( \frac{m_{\text{Sun}}}{m_{\text{star}}} \right)^{3}\text{Years}$$

> If we remember that the Sun's mass is $8 \cdot \text{10}^{\text{30}}$
> Kg, knowing the stellar mass, we can calculate the period in which the
> star burn the Hydrogen.

3.  The star *turn off*: the star exhaust the Hydrogen in core and it
    continues to burn it in shells. to Helium. For a 'short' period the
    stellar temperature decreases without a great change of luminosity.

4.  The star burns Hydrogen in shell (red giants) through the CNO cycle:
    as its radius increases so its luminosity raises.

5.  *Helium flash*: the star begins to burn Helium maintaining the
    combustion of Hydrogen in shells. The stellar temperature increases
    very quickly, because the new reactions give a great quantity of
    energy.

6.  The star crosses a fast period of luminosity's variability
    (*Horizontal branch*) till to reach a new configuration of
    equilibrium during which the star burns Helium in core and Hydrogen
    in shells.

7.  If the stellar mass is not enough to light new thermonuclear
    reaction beyond the Helium burning, it 'died' as a white dwarfs of
    Helium.

In Fig. 1 there aren't stars with mass greater than 7-8 solar mass: the
stars with this mass are able to begin new fusion reactions involving
the burning of Helium, Carbon, Oxygen, Magnesium and Neon. A star with a
mass greater than 10 solar mass can develop thermonuclear reaction until
the Iron, ending its life in a supernova type II.

The *luminosity* of a star is the energy radiated from all its surface
in one second. The *absolute magnitude* (*M*) is used usually to measure
the luminosity of the stars; it is connected to the stellar luminosity
by an approximate relation:

$$M \simeq - 2\text{.}5\text{log}\left( \frac{L_{\text{star}}}{L_{\text{Sun}}} \right) + 5$$

We don't need students have familiarity with the logarithm, since we
will use the inverse formula:

$$\frac{L_{\text{star}}}{L_{\text{Sun}}} \simeq \text{100} \cdot \text{10}^{\left( - \frac{2}{5}M \right)}$$

We remember that Sun's luminosity is:
$L_{\text{Sun}} = 4 \cdot \text{10}^{\text{26}}$ Watt. The stellar
luminosity is connected by the mass through this correlation:

$$L_{\text{Star}}:L_{\text{Sun}} = m_{\text{Star}^{3\text{.}5}}:m_{\text{Sun}^{3\text{.}5}}$$

Using all these easy relationships and the HR diagram, if we know one
parameter, we are able to estimate the rest. For example, the *absolute
magnitude* of Sirius (α Canis Majoris) is *M=+1.5*. With this parameter,
using \[1.2\] we can find $L_{\text{Star}}L_{\text{Sun}} = \text{25}$;
than using \[1.3\] we obtain its mass:
$m_{\alpha\text{CM}} = 2\text{.}4 \cdot m_{\text{Sun}}$. Through a graph
method, as illustrated in Fig. 2, the student can estimate the spectral
class of α Canis Majoris. At the end we can appreciate how long the star
stay into main sequence, using the \[1.1\] the value is 0.7 billions of
years.

![](media/image3.png){width="5.5in" height="3.902755905511811in"}So if
we have the absolute magnitude of a star, connecting all the equations,
we are able to estimate these parameter: luminosity, mass, spectral
class (then temperature) and the time of permanence in the main sequence
of a star. This is a great result that represents a kind of revelation
for young student with fundamental basis of algebra.

**The development in the classroom**

This is only a suggestion about the modality with which is possible to
introduce the arguments: obviously much depends on the student's
background. A helpful instrument could be a Power Point presentation as
we present [[here]{.underline}](../1.ppt).

The lesson can be divided in two part: the first one in which the
teacher show some Power Point slides, just to introduce the argument,
and the second part in which the class is divided in four groups for a
'competition' game.

#### I part

The teacher shows the first Power Point slide and he asks: "What do you
see?". The students are stimulate to answer developing their
observational capacities. A good initial discussion helps the teacher to
understand the wrong ideas of students and to focus the main priors. A
second good question can be: "According to you, what are the main
parameter with which we can study a star?". After the teacher has
collect the classroom's suggestions, he summarizes the fundamental
parameters in the second Power Point slide. It can be useful to present
the third slide which shows the values of these parameter for the Sun.

The fundamental question to ask the student is: "*According to you is
there a correlation between the color and the luminosity of a star?*".
The persons saying "yes" form a group: the teacher gives them have two
dies, one with numbered faces (-10, -5, 0, +5, +10, +15) and one with
colored faces (O, B, A, F, G, K) corresponding to spectral classes.
After 30 throws they obtain a random distribution on the plan (spectral
class Vs absolute magnitude) (Fig. 3). On the other hand the teacher
give a singles die, such that each face contains both magnitude and
color, to the students that believe in a correlation between these two
parameter. This game emphasizes as the correlation between the two
observable appears collecting on the plan an enough number of events.

![](media/image4.png){width="2.654166666666667in"
height="2.479861111111111in"}

Next the teachers presents the three slides about the spectral
classification and HR diagram. The teacher can freely expand the aspects
which he thinks important about evolution, thermonuclear burning (see a
short movie), which are however not essential for our.

The final slide is about the relationship between the main parameters:
after the previous explanations, all the symbols must be evident. The
formulas should not be presented in an ordered way, so that the student
has the time to reflect and to discover the relationships between them

#### II part

The teacher divide the class in four parts, in order to form small
groups of about 5 students. He gives to every group the three carts with
the three formulas, the HR diagram and the sheet with the absolute
magnitude of some important stars (see material for student). We have
chose stars that the boys can recognize in the sky every night. The
challenge is to fill the sheet using the formulas and HR diagram.

This little game stimulate the student to increase the connection
ability and it help them to become familiar with the sizes of the
astrophysics; moreover it incite them to have a collaboration's spirit.
The lesson end with a discussion about the found values, the used method
and the student's curiosity about these arguments.

We suggest to not give them the right value of the star: about it, we
will present a general discussion in the fourth part. It's important to
stimulate the boys with the questions like:

-   What kind of relationship is there between the absolute magnitude
    and luminosity?

-   What kind of relationship is there between the luminosity and the
    stellar mass?

-   What kind of relationship is there between the mass and the time of
    permanence in the main sequence?

# **Material**

# The 'three carts'

-   HR diagram

-   ![](media/image6.png){width="0.9in"
    height="0.6534722222222222in"}Sheet to fill

$$\frac{L_{\text{star}}}{L_{\text{Sun}}} = \text{100} \cdot \text{10}^{\left( - \frac{2}{5}M \right)}$$

![](media/image6.png){width="0.9in" height="0.6534722222222222in"}

$$L_{\text{Star}}:L_{\text{Sun}} = m_{\text{Star}^{3\text{.}5}}:m_{\text{Sun}^{3\text{.}5}}$$

![](media/image6.png){width="0.9in" height="0.6534722222222222in"}

$$t_{m\text{.}s\text{.}} = \text{10}^{\text{10}} \cdot \left( \frac{m_{\text{Sun}}}{m_{\text{star}}} \right)^{3}\text{Years}$$

![](media/image7.png){width="10.5in" height="7.31875in"}

+----------+-----------+----------+-----------+----------+-----------+
| **Star** | *         | $$       | **Mass**  | **       | **Main    |
|          | *Absolute | \frac{L_ |           | Spectral | sequence  |
|          | Ma        | {\text{s |           | class**  | l         |
|          | gnitude** | tar}}}{L |           |          | ifetime** |
|          |           | _{\text{ |           |          |           |
|          |           | Sun}}}$$ |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Sirius   | +1.5      |          |           |          |           |
| (α Canis |           |          |           |          |           |
| Majoris) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Arcturus | -0.3      |          |           |          |           |
|          |           |          |           |          |           |
| (α       |           |          |           |          |           |
| Bootis)  |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Vega(α   | +0.5      |          |           |          |           |
| Lyrae)   |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Capella  | -0.6      |          |           |          |           |
| (α       |           |          |           |          |           |
| Aurigae) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Rigel (β | -8.2      |          |           |          |           |
| Orionis) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Procyon  | +2.7      |          |           |          |           |
| (α Canis |           |          |           |          |           |
| Minoris) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Altair   | +2.4      |          |           |          |           |
| (α       |           |          |           |          |           |
| Aquilae) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| A        | -0.6      |          |           |          |           |
| ldebaran |           |          |           |          |           |
| (α       |           |          |           |          |           |
| Tauri)   |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Antares  | -5.0      |          |           |          |           |
| (α       |           |          |           |          |           |
| Scorpii) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Spica (α | -2.9      |          |           |          |           |
| V        |           |          |           |          |           |
| irginis) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Polaris  | -3.7      |          |           |          |           |
| (α Ursae |           |          |           |          |           |
| Minoris) |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Deneb (α | -6.2      |          |           |          |           |
| Cygni)   |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+
| Regulus  | -0.2      |          |           |          |           |
| (α       |           |          |           |          |           |
| Leonis)  |           |          |           |          |           |
+----------+-----------+----------+-----------+----------+-----------+