How to estimate age of asteroid family Erigon I have one problem:


  On picture below we can see Erigone family asteroids in plane semimajor axis - absolute magnitude. Based on V-type as a consequence of Yarkovsky effect, estimate age of this family bearing in mind that members of this family have spectral of C-type. For average value of change of semimajor axis which belongs to this family take $da=5 \times 10^{-4} \text{AU}$ (for object with diameter $D=1 \,\text{km}$ and time $t=1$ million years). Ignore initial size of family.




I really don't know how to solve this problem (I am studying math, not astronomy). Professor won't tell us how to do it, and I tried to find the answer in his textbook and in two more books in my language, but there is no similar problem at all.

Could you point me in the right direction?
I'm not a professional astronomer, but I'm going to take a stab at this. If anyone sees any problems, feel free to correct them.

The Yarkovsky effect basically means that spinning asteroids over time will be pushed into a further orbit (more precisely, it increases the semimajor axis of the orbit) due to uneven heating and cooling.

Apparently to estimate the age of an asteroid family, using only the Yarkovsky effect, we would start out with this equation:

$$0.2H = \log(\Delta a / C)$$

This equation assumes a fixed geometric albedo for the whole family. $H$ here is the absolute magnitude. $\Delta a$ is equal to $a - a_c$ where $a$ is the semimajor axis.

$$C = \sqrt{p_v} (da/dt)_0 T \cos \epsilon$$

$p_v$ is the geometric albedo.  $(da/dt)_0$ is the maximum Yarkovsky drift rate for a body of size $D_0$, where $D_0$ is an arbitrary reference size (we could use your provided $D$ value above). $T$ is the age of the family that we are trying to calculate. $\epsilon$ is the spin axis obliquity.

Note: I've had to remove my example solution because I could not figure out how to correct my errors from accidentally putting $C$ outside the logarithm.

You can check out more details in my source below.

Source:

"Yarkovsky/YORP chronology of asteroid families" - Vokrouhlickýa, Broža, Bottke, Nesvorný, Morbidelli