<p>Before I attempt the problem, here is a list of assumptions I'm making:</p>
<ul>
<li>This <span class="math-container">$1.3 \cdot 10^6$</span> km value is the distance from Saturn's center.</li>
<li>The probe died at the center of Saturn (<span class="math-container">$r=0$</span>), which is a safe assumption because Saturn's radius is roughly negligible (<span class="math-container">$5.8232\cdot 10^4$</span> km, or 4.45%) compared to the starting distance.</li>
</ul>
<p>First, we need to convert the dates, 12 Sep at 1:27 AM and 15 Sep at 6:32 AM, to seconds: 3 days elapse between these times, and another 5 hours and 5 minutes elapse in the hours. This translates to <span class="math-container">$3\cdot86400 + 5\cdot 3600 + 5\cdot60 = 277500$</span> seconds. However, this is only half the orbital period so we must multiply by 2 to get <span class="math-container">$555000$</span> seconds</p>
<p>Because we have a periapsis of <span class="math-container">$q=0$</span> and apoapsis of <span class="math-container">$Q=1.3 \cdot 10^6$</span> km, <span class="math-container">$a=\frac{Q}{2}=6.5\cdot10^5 \text{ km} = 6.5\cdot10^8 \text{ m}$</span></p>
<p>We can then use Kepler's third law:
<span class="math-container">$$\frac{(6.5\cdot10^8 \text{ m})^3}{(2\cdot277500 \text{ s})^2} = \frac{GM}{4\pi^2}$$</span></p>
<p>We can then isolate M to find <span class="math-container">$M = \boxed{5.264\cdot10^{26}\text{ kg}} \approx 88.224 M_\oplus$</span>.</p>
<p>This is actually pretty close to the true mass of <span class="math-container">$5.683\cdot10^{26}\text{ kg} \approx 95.192 M_\oplus$</span>, corresponding to an error of <span class="math-container">$-7.9\%$</span>.</p>
<p>Hopefully this helps, although in the future, I would suggest showing what you already attempted and any prior research done as Stack Exchange is not a problem-exchange forum.</p>

CC BY-SA 4.0
2023-10-22T17:37:17.100Z
Hello, im so sorry, I couldn't answer till now! Thank you very much for sharing your answer! :) I would just have one question. How do you come to the formula a=Q/2. I mean, I don't quite understand, what Q stands for...
mathgirl752
CC BY-SA 4.0
2023-10-23T20:06:17.123Z
@mathgirl752 if you believe this is the best answer, please feel free to accept it by pressing the check mark under the vote button. Q means apoapsis, I will specify that in an edit.
WarpPrime
CC BY-SA 4.0
2023-10-24T08:12:18.557Z
Yes, of course, thank you!
mathgirl752
CC BY-SA 4.0
2023-10-16T16:42:41.613Z
2023-10-23T20:07:13.687Z
2023-10-23T20:07:13.687Z
WarpPrime
<p>In September 2017, the Cassini 1 mission was nearing its end after 20 years due to fuel shortages. The NASA space probe reached its apoapsis on 12 September at 1:27 AM EDT (time on Saturn) with a distance of about <span class="math-container">$1.3 \cdot 10^6$</span> km from Saturn. There, the probe came to a standstill and then crashed into Saturn. On 15 September at 6:32 a.m. EDT, the last signal from Cassini was registered – shortly after its transmission, the probe probably ignited / died away in the atmosphere of the gas giant.
Estimate the mass of Saturn on the basis of the given data.</p>

CC BY-SA 4.0
2023-10-15T08:29:57.780Z
Wonderful exercise. Any thoughts on your own? What things did you attempt? What are your results and why don't you trust them? Please see https://astronomy.stackexchange.com/help/how-to-ask
I'm kinda hesitant to solve someones homework here without them showing they at least tried anything and what is their particular difficulty in addressing the task... so without any effort on your part shown, I will not be able to answer before at least two weeks passed. I downvoted because that "question does not show any research effort". I will undo that, if you elaborate and show your own attempts
planetmaker
CC BY-SA 4.0
2023-10-15T08:30:57.990Z
This really seems like a homework problem or test question.   What have you tried to get the answer?
notovny
CC BY-SA 4.0
2023-10-15T20:44:47.433Z
Also interesting for an answer: at what level / accuracy do you need one? What type of course is this question asked in?
planetmaker
CC BY-SA 4.0
2023-10-16T14:29:27.740Z
Assume the satellite was on a free fall (no friction). If you know basic equations of motion and Newton's second law you can make an estimate
adrianmcmenamin
CC BY-SA 4.0
2023-10-18T04:54:23.690Z
"There, the probe came to a standstill..." **No it didn't!**
uhoh
<the-sun><mathematics><saturn><space-probe><cassini>
Estimate the mass of Saturn based on the given data below
mathgirl752
# Estimate the mass of Saturn based on the given data below

(Asked by: username_0 on 2023-10-15)

In September 2017, the Cassini 1 mission was nearing its end after 20 years due to fuel shortages. The NASA space probe reached its apoapsis on 12 September at 1:27 AM EDT (time on Saturn) with a distance of about $1.3 \cdot 10^6$ km from Saturn. There, the probe came to a standstill and then crashed into Saturn. On 15 September at 6:32 a.m. EDT, the last signal from Cassini was registered – shortly after its transmission, the probe probably ignited / died away in the atmosphere of the gas giant.
Estimate the mass of Saturn on the basis of the given data.

**Tags:** the-sun, mathematics, saturn, space-probe, cassini

> **Comments:**
> **username_1**: Wonderful exercise. Any thoughts on your own? What things did you attempt? What are your results and why don't you trust them? Please see https://astronomy.stackexchange.com/help/how-to-ask I'm kinda hesitant to solve someones homework here without them showing they at least tried anything and what is their particular difficulty in addressing the task... so without any effort on your part shown, I will not be able to answer before at least two weeks passed. I downvoted because that "question does not show any research effort". I will undo that, if you elaborate and show your own attempts  
> **username_2**: This really seems like a homework problem or test question.   What have you tried to get the answer?  
> **username_1**: Also interesting for an answer: at what level / accuracy do you need one? What type of course is this question asked in?  
> **username_3**: Assume the satellite was on a free fall (no friction). If you know basic equations of motion and Newton's second law you can make an estimate  
> **username_4**: "There, the probe came to a standstill..." **No it didn't!**  

---

## 1 answers

### Answer by username_5 on 2023-10-16. Score: 3 | Accepted answer

Before I attempt the problem, here is a list of assumptions I'm making:

* This $1.3 \cdot 10^6$ km value is the distance from Saturn's center.
* The probe died at the center of Saturn ($r=0$), which is a safe assumption because Saturn's radius is roughly negligible ($5.8232\cdot 10^4$ km, or 4.45%) compared to the starting distance.

First, we need to convert the dates, 12 Sep at 1:27 AM and 15 Sep at 6:32 AM, to seconds: 3 days elapse between these times, and another 5 hours and 5 minutes elapse in the hours. This translates to $3\cdot86400 + 5\cdot 3600 + 5\cdot60 = 277500$ seconds. However, this is only half the orbital period so we must multiply by 2 to get $555000$ seconds

Because we have a periapsis of $q=0$ and apoapsis of $Q=1.3 \cdot 10^6$ km, $a=\frac{Q}{2}=6.5\cdot10^5 \text{ km} = 6.5\cdot10^8 \text{ m}$

We can then use Kepler's third law:
$$\frac{(6.5\cdot10^8 \text{ m})^3}{(2\cdot277500 \text{ s})^2} = \frac{GM}{4\pi^2}$$

We can then isolate M to find $M = \boxed{5.264\cdot10^{26}\text{ kg}} \approx 88.224 M\_\oplus$.

This is actually pretty close to the true mass of $5.683\cdot10^{26}\text{ kg} \approx 95.192 M\_\oplus$, corresponding to an error of $-7.9\%$.

Hopefully this helps, although in the future, I would suggest showing what you already attempted and any prior research done as Stack Exchange is not a problem-exchange forum.

> **Comments:**
> **username_0**: Hello, im so sorry, I couldn't answer till now! Thank you very much for sharing your answer! :) I would just have one question. How do you come to the formula a=Q/2. I mean, I don't quite understand, what Q stands for...  
> **username_5**: @username_0 if you believe this is the best answer, please feel free to accept it by pressing the check mark under the vote button. Q means apoapsis, I will specify that in an edit.  
> **username_0**: Yes, of course, thank you!  

---

