# Are there any equations out there that can calculate the burnup of an asteroid depending on angle?

Are there any equations that can calculate the burnup of an asteroid depending on the angle? I want to be able to calculate the magnitude of an asteroid impact with the angle being the changeable variable

• This is unlikely, as you never know the consistency or makeup of the asteroid. There are indicative numbers from observed data, which may be enough for you to fit an equation to... Jul 10, 2019 at 19:10
• I am looking into the asteroid Apophis in which we know the makeup of Jul 11, 2019 at 5:10

## 1 Answer

The Purdue University Impact Earth! site will calculate this for you. Clicking on the Tell me more link will give you the paper of formula and approximations that are used. The equation for the breakup altitude is: $$z_\star\approx-H\Bigl[\ln\Bigl(\frac{Y_i}{\rho_0v_i^2}\Bigr)+1.308-0.314I_f-1.303\sqrt{1-I_f}\Bigr]$$ where $$H$$ is the scale height of the atmosphere (taken to be 8km), $$\rho_0$$ is the surface atmospheric density (taken to be $$1\,kg/m^3$$) and $$Y_i$$ is the yield strength of the impactor and $$I_f$$ are given by: $$\log_{10}Y_i=2.107+0.0624\sqrt{\rho_i}$$ where $$\rho_i$$ is the impactor density (2600 kg m$$^{-3}$$ for Apophis (assumed)) and: $$I_f=4.07\frac{C_DHY_i}{\rho_iL_0v_i^2\sin\theta}$$ here $$C_D$$ is the drag coefficient (taken to be 2), $$L_0$$ is the impactor diameter (0.34 km for Apophis; Brozovic et al. 2018), $$v_i$$ is the impact velocity and $$\theta$$ is the entry angle. If $$I_f<1$$, this indicates the impactor breaks up above the surface.

Given the complexities involved, I would suggest you use the website itself and vary the impact angle through tha - the dominant effects are going to be the encounter velocity (see e.g. close approach data at the JPL Small Body DB Browser) and uncertainties in the size and density.

• Great, right out of Collins, Melosh & Marcus 2005, but you made several mistakes and did not state what I sub f is (If). vi is not the impact velocity, that is vo. vi is the initial velocity. But what is "If" I know what it does (influences the breakup altitude z*) and the equation is easy to check, but what is it? Mar 16 at 13:54
• I didn't provide a link-only answer as those are bad, particularly as in this case, the link moves... The authors don't define $I_f$ either; it appears to be some form of "impactor fraction" of how much hits the surface and forms the crater; quote: "the impactor may reach the surface intact; in this case, $I_f >1$". $v_i$ is velocity at breakup/impact though; paper says "The properly decremented velocity, calculated using Equation 8 is used for crater size" and that eqn gives velocity after atmospheric drag is in terms of $v_0$ which is stated to be 'velocity at top of the atmosphere' Mar 16 at 19:35