How to get a plot for an equation

Questions
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1

Hello, I’m trying to code the equation shown below and get a curve or possible solution as shown below as well (not all of them). This is my attempt but not sure if it is correct and how to obtain a similar plot?. Are the units right? Any help will be appreciated. Thanks

ClearAll["Global`*"];
T = Quantity[1.2 * 10^6, "Kelvins"];
kb = Quantity[1, "BoltzmannConstant"];
Ms = Quantity[1.99 *10^30, "Kilograms"];
G = Quantity[6.67 *10^-11, "Meters^2*Newton/Kilograms^2"]; 
mH = Quantity[1.99 * 10^30, "Kilograms"];
r = Quantity[1*10^9, "Meters"]
u = Quantity[17, "Meters"/"Second"]

    uc = Sqrt[2*kb*T/mH]
    rc = G*Ms*mH/4*kb*T
    eq = (u/uc)^2 - 2 * Log[u/uc] == 4 *Log[r/rc] + 4 *(rc/r) - 3

Screenshot2023-11-02at2.24.22 PM.png
Screenshot2023-11-02at2.24.22 PM.png792×128 9.36 KB

Screenshot2023-11-02at12.32.11 PM.png
Screenshot2023-11-02at12.32.11 PM.png924×532 25.9 KB

2

If you take the logarithm, as in Log[r/rc], the argument must be dimensionless. In your case r is a length but rc is not.

3

Agreed, there must be something wrong with your code for uc and rc. Can you post the original equations for those?

4

These are the equations for rc and uc. Any help will be appreciated. Thank you

Screenshot2023-11-05at6.46.49 AM.png
Screenshot2023-11-05at6.46.49 AM.png284×266 6.85 KB

5

So as Gianluca points out, u and u_c have the same units (m/s), but r and r_c do not. r is in meters, but r_c based on your code is in newtons * seconds^2 / kilograms. Could there be something wrong with your definitions of Ms and mH? You have the both as (identical) masses. But the different capitalization in your source suggests they might not be the same thing?

6

Ms ans mH are different things. If rc, r, u and uc are assumed to be dimensionless, is there any way to get a curve like the ones shown in the previous plot? Thanks

7

Well that’s odd because they are the same in your code. Perhaps you can post the reference for these equations? Something to do with gas particle velocity I assume…

8

Thanks for your reply. This is the reference.

DYNAMICS OF THE INTERPLANETARY GAS AND MAGNETIC FIELDS

9

I see that Figure 1 is what you are trying to recreate, but the relevant equations (16 and 17) have a different form from the ones you posted earlier and, alas, I don’t have the time to try to see how your version relates to those in the article. Especially as I’m not a physicist!