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#### n3tjm

##### Papa Elf
Yesterday I was working on some more house cleaning when I got a visitor . This Dark Gray Tabby used to belong to a neighbor that left him behind when they moved. I see him in my yard once in a while.

He likes it when I pet him and scritch him. He likes rolling around in the dirt. Oh, when he sees me outside he meows and walks up to me. and he likes to rub himself on my legs, and on my steps, and door, and side of my building.
I think he wants to move in with me . Unfortunately there is a no pets rule in my apartment building, so if it wasn't for that I would of adopted him

I wondered if he still had his claws. He does. He was showing off himself he was stretching and I saw his claws. Its good that he has them if he is living on the street. There are mean kitties out there.

#### shrox

##### Well-Known Member
Let him in, just say you keep leaving that darn door open...

#### sunward

TRF Supporter
Those eyes. How could you refuse?

He is hypnotizing you to take him in. Be careful.

#### n5wd

##### Well-Known Member
...used to belong to a neighbor that left him behind when they moved.
I hope there's a special penalty box for people that abandon domesticated animals instad of arranging care for them! Is anyone feeding him? If not, you might check with the local no-kill animal shelter to see whether they can help you trap him. He's got definite adoption potential, and letting him roam free is hazardous to his health.

#### gerbs4me

##### Well-Known Member
The cat looks beautiful, needs a good home, too bad there is the no pet rule at your apartment.

#### shrox

##### Well-Known Member
Kitty has already picked his new home.

Last edited:

#### Mikus

##### Well-Known Member
I hope there's a special penalty box for people that abandon domesticated animals instad of arranging care for them!
Seconded.

#### MaxQ

##### Tripoli 2747
I have three strays living in my back yard....same deal, left by renters in the neighborhood. Since we have two cats indoors we can't have them inside.
We feed and water them twice a day and have an insulated kitty condo set up outside in the winter.

One is a real tough guy...orange tabby we named Bob.

We noticed one summer that we hadn't seen Bob in days...then I saw an object
jumping up and down in the garage window...it was Bob.
He had been in there several days....no food or water....but nine lives.
He got in my garage out back while I was mowing the lawn...

lot of respect for this guy Bob...

#### troj

##### Wielder Of the Skillet Of Harsh Discipline, Potent
I hope there's a special penalty box for people that abandon domesticated animals instad of arranging care for them!
Agreed!

But then, my cats don't ever go outside, unless it's in a carrier, or being carried, on their way to the vet.

-Kevin (who has far too many very spoiled animals in his house)

#### dave carver

##### ....what hump?
I have outdoors kitties. Started with a scrawny, half starved, half kitten Calico yowler we called Screech.

I fed her a can of tuna to the sounds of nom nom hisss nom nom hisss then a few days later it was nom hiss nom meow nom hiss nom meow until finally it was nom meow nom meow nom nom. She had a series of kittens that unfortunatly have dissapeared over the years, one a Tuxedo we called Groucho because of his moustach, he was getting to be a BIG kitty then one day he just wasn't there anymore

We still have Screech's kitten Ditto(called that because she looks just like mama), two yellows that were Screech's last kittens and Ditto's FOUR little one's. I'm on a farm and I can tell you they earn their keep in cat food by keeping down the mice, big time. Used to get invaded every winter but with the cats nary a one for the last 4 years.

I assembled a cat house from carpet and carpet padding that they go into for shelter, it's very insulated and when it's going to get real cold I turn a 75 watts bulb on under the house to keep it warmer.

They, like I said, are outdoors kitties, allergies keep me from having any indoors so as I tell people Kitties come and go around here. I wish it were different but with the allergies I have no choice except drugs and that's not a option because of other drugs I have to take.

But I really like kitties

#### tquigg

##### Well-Known Member
We have a big, old, fat calico cat named "Mama Kitty" that was left behind by her "druggie" owner who lived next door to us. She's a real love, but prefers to stay outdoors.

We have a another cat named Rhubarb, or "rudy" for short. My son found him as a small kitten abandoned under the leaves of a rhubarb plant in the neighbor's garden, and he's been with us every since.

Recently, a cat abandoned her six kittens in a basement window well at our house. One died, but we nurtured the other five. We kept the runt of the litter, named him "Squirt". He looks like he has a lot of Oriental in him. He's a real love, and the vet gave him a clean bill of health, all shots, etc. We raised his brothers and sisters, and have found loving homes for all of them.

A few months ago, as I was backing my car out to go to work, I found two small kittens huddled under the front of the car, Nearly ran over both of them! Swooped and scooped them, and have raised them up to be real loves as well. We've found homes for them, too.

You have to love youself before you can love animals.

#### n3tjm

##### Papa Elf
Heard another story about this cat. They said that the owner did take him but he keeps jumping out the window and coming back here? I personally dont buy that story

##### Roger Smith
I hope there's a special penalty box for people that abandon domesticated animals instad of arranging care for them!
A neighbor's cat got pregnant and the neighbor locked it out of the house until after it had the babies. So, we ended up with several strays living in our backyard. We tried to catch them to get them vaccinated, "fixed," and, hopefully, adopted.

But, it's not easy catching stray cats and a second generation was born before we finally got them all spayed and neutered. Eventually we found homes for four of them and took one in ourselves.

The remaining four are still strays, but we feed them in our back yard and they are fixed.

-- Roger

##### Roger Smith
A few months ago, as I was backing my car out to go to work, I found two small kittens huddled under the front of the car, Nearly ran over both of them!
One of the strays living in our back yard must have crawled under the hood of a car and had its tail cut off. It took a while for us to catch her. She really looked bad when we took her to the vet. But, they cleaned her up, fixed up what was left of her tail, spayed her and gave her vaccinations. Now, she's not the prettiest cat in the world, but she's healthy.

One of the strays we tried to help, however, had to be put down. It was just too sick. We must have spent at least $500 out of our pockets to help these cats and prevent them from having more kittens. (Not to mention the suffering some of the cats have endured.) All because one irresponsible neighbor didn't have his cat fixed and let it roam outside. -- Roger #### shrox ##### Well-Known Member Looking at kitty's face make me think it's not a house cat, it's a house panther. #### ben_ullman ##### Well-Known Member I have three strays living in my back yard....same deal, left by renters in the neighborhood. Since we have two cats indoors we can't have them inside. We feed and water them twice a day and have an insulated kitty condo set up outside in the winter. One is a real tough guy...orange tabby we named Bob. We noticed one summer that we hadn't seen Bob in days...then I saw an object jumping up and down in the garage window...it was Bob. He had been in there several days....no food or water....but nine lives. He got in my garage out back while I was mowing the lawn... lot of respect for this guy Bob... does his last name happen to be Utley? :roll: Ben #### luke strawwalker ##### Well-Known Member Seconded. Yeah... I heard that... You guys should try living in the country... everybody seems to think that we're just here to adopt their critters once the kids outgrow them, they move, or just don't want them anymore. We end up with more strays running around than you can shake a stick at! We had some renters move out of the old farmhouse at Shiner, and they had a PACK of dogs (like 4-5) and were getting the last load of junk out of the house (after not having paid the rent for a couple months) and judging by how full the truck was, had NO intention of taking the dogs. I told them they WOULD be taking their dogs, because stray dogs on a cattle ranch start killing calves when they get hungry enough, and I already had some suspicions that a calf or two that we lost was because of their dogs. They grudgingly loaded them up and left with the last of their junk. After we finished working on the farm that day, I locked up the gates and decided to go the back way to Hallettsville and stop by the sheriff's office to request a frequent patrol by the farm, since this jerk didn't exactly leave under 'the best of circumstances' and he wasn't what I'd call a 'model citizen'. Halfway to Hallettsville I find his pack of dogs, roaming the countryside by the road out between farms. When I requested the frequent patrol, I informed the deputy that the guy had dumped his dogs on the road halfway back to the farm, and he might want to dispatch animal control out there to pick them up (I worked with animal control in Wharton when I was in the police academy back in 96 and this wasn't too long after that) and he looked at me funny and said, "well, we don't have animal control in Lavaca County, but yeah I'm gonna go have a talk with him about those dogs". I, being a cattleman, asked, "well, what do you do when farmers call with complaints about dogs chasing or molesting their livestock??" He looked a bit sheepish and said, "well, folks just have to handle it themselves, as best they see fit." We've had renters with dogs that chased cows in the pond, one cow drowned from a dog chasing her in and not letting her come out, and I think that's the only time Grandpa had to shoot a dog, and he hated to do it, but the law allows for livestock farmers to protect their animals, and he told the renter several times to chain the dog or give it away. It's a mess... OL JR #### luke strawwalker ##### Well-Known Member We have a big, old, fat calico cat named "Mama Kitty" that was left behind by her "druggie" owner who lived next door to us. She's a real love, but prefers to stay outdoors. We have a another cat named Rhubarb, or "rudy" for short. My son found him as a small kitten abandoned under the leaves of a rhubarb plant in the neighbor's garden, and he's been with us every since. Recently, a cat abandoned her six kittens in a basement window well at our house. One died, but we nurtured the other five. We kept the runt of the litter, named him "Squirt". He looks like he has a lot of Oriental in him. He's a real love, and the vet gave him a clean bill of health, all shots, etc. We raised his brothers and sisters, and have found loving homes for all of them. A few months ago, as I was backing my car out to go to work, I found two small kittens huddled under the front of the car, Nearly ran over both of them! Swooped and scooped them, and have raised them up to be real loves as well. We've found homes for them, too. You have to love youself before you can love animals. I had something similar happen... darn cats can be SO goofy at times! I drive a schoolbus and so I leave WELL before dawn, and I get in the truck and take off, dropping my daughter at my parent's house to watch her while I drive the bus. I get back to the school after the route, park and check the bus, and go out to the parking lot to head home, and from ten feet away I hear my truck "meowing"... I go around the truck trying to figure out where it's coming from, and lo and behold it's from behind the grill/front bumper area. I pop the hood and look down between the radiator support and grill, and turns out one of the goofy yard cats had moved her kittens into the truck, right in front of the radiator on the rubber flap connecting the bumper and radiator support, to funnel the air from the grill through the radiator. SO, I had to reach through the tow hook holes in the bumper on either side while trying to catch the little hissing nippers and pull them out, while looking through the slot between the radiator support and grill. Finally caught them all and they rode home in the truck. I get back to the house and the momma cat comes charging out meowing loudly and hissing at me, as I fetch her babies and put them on the porch, and I looked at her and told her, "Don't hiss at me; YOU were the moron that put them under the pickup!" She prowled around a bit and finally took them elsewhere under the house someplace. Dad left for work one evening and called me to check on something that was in the bottom of the dry mudhole under his car when he left. I went down and found a runt but otherwise healthy baby raccoon abandoned there, so I took him home and built a cage for him, and raised him. Goofiest little thing-- he ate cat food, which I'd set in an old coffee cup twice a day into his cage for him, along with a big water bowl, and the silly thing would sit there and sift through the entire cup of cat food looking for the biggest piece, take it out, wash it in the water with his hands, and then eat it, and then sift for the next largest piece, wash it, eat it, and then look for the third largest piece, until he'd eaten the whole cup of cat food that way. Then he'd drink some water, which by this time was more like gravy, so I'd grab a bucket and refill his water bowl right quick. He was pretty good until he got grown and got breeding age, then they get mean. I turned him out but he hung around for several weeks, fiddling in the shop when I was welding, picking up bits of metal despite me fussing at him that he was gonna pick up a red-hot one if he wasn't careful-- he'd just look at me sideways and growl a bit. One day I was working and kept hearing noises, and he was in the shop sitting on shelf sifting through a jar of old cotter pins for some reason, and I fussed at him and told him to leave them alone and quit making messes, and he'd turn his head and growl a bit, and amble off. He finally started being gone all day, except coming in morning and evening when I fed the cats, to get his 'share' of the cat food. Finally he left and didn't come back. Somewhere out there is a VERY FAT RACCOON!!! OL JR #### o1d_dude ##### 'I battle gravity' TRF Supporter Based on the pictures and the behavior described, YOU are his new pet human. The difference between dogs and cats is dogs think you are a god and they worship you. Cats on the other hand think they are gods and allow you to worship them. #### Rocketcrab ##### Well-Known Member Take him in - how often does the landlord check on these things? I know, easy for me to say. True story - I had never had anything against cats, I was just allergic to them. In 1996, the summer I turned 40, our then-4 year old daughter started dragging home stray kittens - we live in a housing development, right across the road from the farm that once encompassed what is now houses, and people regularly abandon litters of unwanted kittens there. Well, anyone who has a daughter knows the drill - "PUHLEEEESE Daddy!!! I'll take care of him/her/it!!!" So, I relented and told her that the kitten she dragged home this one particular day could stay, BUT it had to remain outside, because "cats make Daddy swell up like a bullfrog honey." Funny thing - turning 40 cured my allergy. Beats me how or why it happened, but I didn't swell up like a bullfrog. So the kitten moved in to the house. It's still here, 13 years later, along with five [yes, 5] others, making six. All speyed or nutered. The total was up to 7 at one point, but Coleman [he was 100% black] kinda became road pizza this past summer. My favorite is a 9 -year old Maine Coon cat that weighs about 17 pounds, thinks he's a dog, and snores louder than I do. He's never really had a name - although I've always called him "Elvis" because he moved in on the anniversary of The King's death. Our vet bills now approach or human medical bills. But the darn things do grow on you, I have to admit. Akll this to say N3TJM, I know where you're coming from - you want to help the poor thing. Last edited: #### Pantherjon ##### Well-Known Member Based on the pictures and the behavior described, YOU are his new pet human. The difference between dogs and cats is dogs think you are a god and they worship you. Cats on the other hand think they are gods and allow you to worship them. Yep, looks like the cat has adopted you! I bit different spin on cat-dog differences; The difference between dogs and cats are that dogs are part of the family, your papa or mama to them, for cats they think they are royalty and you are their servants!:roll: #### JRThro ##### Lifetime Supporter TRF Lifetime Supporter TRF Supporter 5 or 6 years ago, my wife got two kittens, a black shorthair our kids named Blackie and her sister, a gray tabby they named Bounce. They were indoor kittens, but once they got older, they became indoor/outdoor cats. Because we were lousy cat owners, they each had several litters, a few of which we were able to find homes for, but most of which we took to the Humane Society. We also finally took Bounce there. :blush: When things finally settled out a couple of years ago, we still had Blackie and two of her kittens, a black female named Trouble and a gray and white male named Pipsqueak. All three of them have been spayed/neutered and have their shots. They are all indoor/outdoor cats. A couple of years ago, one of our neighbors moved and left behind their black and white male cat named Cowboy (because they "couldn't catch him"), who is a big, muscular cat who is more than willing to get into a fight with just about anything that crosses him. He started showing up at our back door, dirty, scrawny, and sometimes all scratched up. We fed and watered him pretty regularly, but he would never come inside or let anyone get near him. As time went on, he gradually would let my wife get closer and finally pet him and even pick him up and bring him inside. He is one of the biggest house cats I have ever seen. Several months ago, he disappeared for a while, and when he showed up again, he had a huge infected wound and an abscess almost the size of a golf ball on the side of his head, plus various other scrapes and cuts. After several days and several attempts, my wife was able to catch him in a bath towel (to keep from getting scratched and bitten) and put him in our cat carrier to take him to the vet. After surgery, antibiotics, a cage to keep him in outside, neutering, and$400+ later, my wife spent a couple of weeks flushing his wound with saline, giving him oral antibiotics, and taking care of the drain tube on the side of his head until he was all healed up. We're sure he would have died from the infection if she hadn't done anything.

To make a long story short (too late! much too late!), we now have 4 indoor/outdoor cats. Cowboy doesn't get along with the other 3, but they usually just try to avoid each other except for the occasional growl here and there and the fairly rare screaming catfight in our kitchen, usually in the middle of the night.

Also, roughly around the time that Cowboy started hanging around our back door, another cat, a big dark smoky-gray male that our kids have named Tobacco, also started hanging around outside. We now feed him pretty regularly, and he comes inside to eat sometimes. If we let him come inside, he will plant himself in front of the cat feeder and eat until we shoo him back outside, though there have been days when we forget that he's there and my wife will come home to find him sunning himself on a window sill like he owns the place. :shock:

So we have 4-1/2 housecats these days. I never thought I liked cats, dogs, or any other mammalian pets, but it is nice to have Cowboy come up to me and push his head into my hand so I'll rub/scratch/pet him. He also will sit in the chair next to me when I'm on the computer. So yeah, I like him and our other cats. More than I'll admit.

Now if I can just get somebody else to change the litter boxes!

#### MaxQ

##### Tripoli 2747
does his last name happen to be Utley? :roll:

Ben
He isn't anywhere as nice as that Bob....

#### ben_ullman

##### Well-Known Member
He isn't anywhere as nice as that Bob....

Ben

#### RandyM

##### Well-Known Member
I have a simular story. Had a grey tiger start coming by the desk sliding door a few weeks ago. He looks looked like a house cat. My son saw him and tole me we had a stray cats hanging around, I asked him is his name was Brian (as in Brian Setzer, lead guitar in the Stray Cats Band).

After hanging around for a few weeks with us feeding him, we took him in this weekend. He seems to be really happy with a new home. Can't say the same for one of our older cats! He seems to be a little PO'ed about it!

Took him to the vet and it was confirmed that he is in good health. He's now got all of his shots and he is already fixed.

I'll post a pic tomorrow.

#### RandyM

##### Well-Known Member
Here's a pic with my daughter. As you can see, he will find no shortage of attention here!

#### luke strawwalker

##### Well-Known Member
(snip) After several days and several attempts, my wife was able to catch him in a bath towel (to keep from getting scratched and bitten) and put him in our cat carrier to take him to the vet. (snip) :
Ya might want to think through that one again next time...

When I was in the police academy, I worked with animal control as part of my 'inservice' ride-along type stuff, which we had to do a lot of. Part of my job was assisting when we had to put some animals down. The vet administered the shots, but the animal control officer had to 'secure the animals' for the vet to euthanize. I mostly moved cages around and did disposal duty.

Later! OL JR

#### troj

##### Wielder Of the Skillet Of Harsh Discipline, Potent
I never thought I liked cats, dogs, or any other mammalian pets, but it is nice to have Cowboy come up to me and push his head into my hand so I'll rub/scratch/pet him. He also will sit in the chair next to me when I'm on the computer.
Ah, the blessing and the curse of cats!

Having them want attention is great....except when they want attention. NOW. And you're trying to do something else.

It's really, Really, REALLY hard to type when a cat is in your lap and head-butting your hand up off of the keyboard, because that hand is supposed to be giving said cat head scratches. NOW! Yes, NOW.

A dog will come up, give you a forlorn look, look at you with pitiful eyes, and hope it gets attention. A cat comes up and demands attention. NOW, you stupid human who has no other purpose than to serve me.

-Kevin (two cats, two dogs)

#### Donaldsrockets

##### Well-Known Member
Ah, the blessing and the curse of cats!

Having them want attention is great....except when they want attention. NOW.
This describes the two cats I have to a T.

They're both nearly 12 years old and showing absolutely NO signs of slowing down yet.

I do love them though, I wouldn't trade them for anything.

Update on kitty?


text:
ArticlePDF Available

# Time-evolution of the ion velocity distribution function in the discharge of a Hall effect thruster

Authors:

## Abstract and Figures

The temporal characteristics of the Xe$^+$ ion axial Velocity Distribution Function (VDF) were recorded in the course of low-frequency discharge current oscillations ($\sim$~14 kHz) of the 5 kW-class PPS$\circledR$X000 Hall thruster. The evolution in time of the ion axial velocity component is monitored by means of a laser induced fluorescence diagnostic tool with a time resolution of 100 ns. As the number of fluorescence photons is very low during such a short time period, a hom-made pulse-counting lock-in system was used to perform real-time discrimination between background photons and fluorescence photons. The evolution in time of the ion VDF was observed at three locations along the thruster channel axis after a fast shut down of the thruster power. The anode discharge current is switched off at 2 kHz during 5 $\mu$s without any synchronization with the current oscillation cycle. This approach allows to examine the temporal behavior of the ion VDF during decay and ignition of the discharge as well as during forced and natural plasma oscillations. Measurements show that the distribution function of the axial component of the Xe$^+$ ion does change periodically in time with a frequency close to the current oscillation frequency in both forced and natural cases. The ion density and the mean velocity are found to oscillate whereas the velocity dispersion stays constant, which indicates that ionization and acceleration layers have identical dynamics. Finally, variations over time of the electric field are for the first time experimentally evidenced in a crossed-field discharge.
Content may be subject to copyright.
arXiv:0902.1432v1 [physics.plasm-ph] 9 Feb 2009
Time-evolution of the ion velocity distribution function in the
discharge of a Hall eﬀect thruster
S. Mazouﬀre, D. Gawron, N. Sadeghi,
February 9, 2009
ICARE, CNRS, 1C avenue de la Recherche Scientiﬁque, 45071 Orl´eans, France.
LSP, Joseph Fourier University - CNRS, 140 Av. de la Physique, 38402 St Martin d’H`eres, France.
Abstract
The temporal characteristics of the Xe+ion axial Velocity Distribution Function
(VDF) were recorded in the course of low-frequency discharge current oscillations (14 kHz)
of the 5 kW-class PPSrX000 Hall thruster. The evolution in time of the ion axial veloc-
ity component is monitored by means of a laser induced ﬂuorescence diagnostic tool with
a time resolution of 100 ns. As the number of ﬂuorescence photons is very low during
such a short time period, a hom-made pulse-counting lock-in system was used to perform
real-time discrimination between background photons and ﬂuorescence photons. The evo-
lution in time of the ion VDF was observed at three locations along the thruster channel
axis after a fast shut down of the thruster power. The anode discharge current is switched
oﬀ at 2 kHz during 5 µs without any synchronization with the current oscillation cycle.
This approach allows to examine the temporal behavior of the ion VDF during decay and
ignition of the discharge as well as during forced and natural plasma oscillations. Mea-
surements show that the distribution function of the axial component of the Xe+ion does
change periodically in time with a frequency close to the current oscillation frequency in
both forced and natural cases. The ion density and the mean velocity are found to oscil-
late whereas the velocity dispersion stays constant, which indicates that ionization and
acceleration layers have identical dynamics. Finally, variations over time of the electric
ﬁeld are for the ﬁrst time experimentally evidenced in a crossed-ﬁeld discharge.
Submitted to Physics of Plasmas
1
1 Introduction
A Hall Eﬀect Thruster (HET) is a gridless ion accelerator that ﬁnds applications in the ﬁeld
of spacecraft propulsion [1, 2]. Such a type of electric propulsive device is especially suited for
long duration missions and for maneuvers that require a large velocity increment. HETs are
at present mostly employed for geostationary communication satellite orbit correction and
station keeping. Other ﬁelds of application are envisaged for the near future. Low power
Hall thrusters seem suited for drag compensation of observation satellites that operate on a
low-altitude Earth orbit. The use of high power Hall thrusters for orbit raising and orbit
topping maneuvers of communication satellites would oﬀer signiﬁcant beneﬁts in terms of
launch mass, payload mass and operational life. In addition, large Hall thrusters appear as
good candidates to be used as the primary propulsion engine for robotic space probes during
interplanetary journeys towards far-oﬀ planets and asteroids.
A Hall eﬀect thruster is a low-pressure DC discharge in crossed electric and magnetic ﬁelds
conﬁguration [2, 3, 4]. Xenon is generally used as a propellant gas due to its high atomic mass
and low ionization energy. A schematic of a HET is depicted in Fig. 1. The anode is located
at the upstream end of a coaxial annular dielectric channel that conﬁnes the plasma. The
cathode is situated outside. A set of coils combined with magnetic parts provide a radially
directed magnetic ﬁeld Bof which the strength is maximum in the vicinity of the channel
exhaust. The magnetic ﬁeld is chosen strong enough to make the electron Larmor radius much
smaller than the channel characteristic sizes, but weak enough not to aﬀect ion trajectories.
The gas injected through the anode is ionized inside the channel by electron impacts. As the
magnetic ﬁeld considerably slows down the electron motion towards the anode, the applied
potential concentrates in a restricted area at the channel entrance. The corresponding axial
electric ﬁeld Ethen accelerates ions out of the channel, which generates thrust. The ion
beam is neutralized by a fraction of electrons emitted from the cathode. The crossed Eand
Bgeometry is at the origin of a large electron azimuthal drift – the Hall current – that is
responsible for the eﬃcient ionization of the supplied gas. When operating near 1.5 kW, a
HET ejects ions at 20 km s1and generates 100 mN of thrust with an overall eﬃciency of
It is well-established that the crossed-ﬁelds discharge of a Hall eﬀect thruster is strongly
non-stationary [3, 6]. This speciﬁc type of magnetized plasma displays numerous types of
oscillations, which encompass many kinds of physical phenomena, each with its own length and
time scales [6]. Current and plasma ﬂuctuations, of which the frequency range stretches from
10 kHz up to 100 MHz, play a major role in ionization, particle diﬀusion and acceleration
processes. Low-frequency plasma oscillations in the range 10-30 kHz, so-called breathing
oscillations, are especially of interest as they carry a large part of the power [7]. Breathing
oscillations ﬁnd their origin in a prey-predator type mechanism between atoms and ions
as shown by Boeuf and co-workers [8]. In short, these oscillations originate in a periodic
depletion and replenishment of the neutrals near the exhaust of the thruster channel due to
the eﬃcient ionization of the gas. The frequency is then linked to the time it takes for atoms
to ﬁll in the ionization region. With an atom thermal speed of 300 m/s and a region size of
20 mm, one ﬁnds a frequency of 15 kHz. The breathing phenomenon not only disturbs the
discharge current but also has a strong impact on several quantities. The plume shape and
the ion beam divergence change during an oscillation cycle of the current as was shown by
means of CCD imaging [9]. A time variation of the ion beam energy in a Hall thruster far
ﬁeld was evidenced using a retarding potential analyzer [11]. The electron density and the
2
plasma potential oscillate at low frequency [10]. Conversely, the electron temperature stays
unchanged. Oscillations of the aforementioned quantities are most likely connected with a
time variation of the potential distribution or, in other words, with the variation with time
of the accelerating electric ﬁeld. Therefore, it appears of great interest to investigate the
temporal behavior of the electric ﬁeld that certainly hides a rich and intricate dynamics.
Across the acceleration layer, the medium is collisionless, i.e. ion-ion and ion-atom collision
events are scarce. As a consequence, the electric ﬁeld can be directly inferred from the Xe+
velocity which then becomes the quantity to be examined.
Laser Induced Fluorescence (LIF) spectroscopy in the near infrared has often been used in
the past few years to measure the time-averaged velocity distribution function (VDF) of Xe+
ions in the plasma of a Hall eﬀect thruster [12, 13, 14, 15]. LIF spectroscopy is a non intrusive
diagnostic tool that enables an accurate determination of the local velocity of atoms along the
laser beam direction by measuring the Doppler shift of absorbed photons. The metastable
Xe+ion VDF is recorded by collecting ﬂuorescence radiation at 541.9 nm after excitation
of the 5d 2F7/26p 2Do
5/2transition at λ= 834.7233 nm [14]. A phase sensitive detection
method is often used to capture the ﬂuorescence signal. However, this method, which is
powerful enough for the extraction of a signal in an environment with a high background
noise level, oﬀers a poor time resolution. To achieve the measurement of the time-resolved
Xe+ion VDF in the plasma of a Hall eﬀect thruster, it is necessary to develop a bench able
to detect LIF photons with a time resolution around 1µs. In normal operating conditions
for a HET, the number of ﬂuorescence photons observed at 541.9 nm with a continuous laser
beam tuned at 834.7233 nm with about 1 mW/mm2power density is on the order of 102per
µs. Under identical experimental conditions, the number of background photons generated
by the plasma at 541.9 nm during 1 µs is typically 1, which means a ratio of 100 between the
two signal amplitudes. The laser system must therefore be able (i) to detect a tiny amount
of photons hidden in a strong background (ii) to determine with a high accuracy the exact
moment in time ﬂuorescence photons have been produced. One must therefore turn to a
photon-counting technique.
In this contribution, we present time-resolved measurements of the Xe+ion axial VDF in
the discharge of the 5 kW-class PPSrX000 Hall thruster ﬁred at 500 V discharge voltage and
6 mg/s xenon mass ﬂow rate. The evolution in time of the VDF was recorded at several loca-
tions during the transient regime that follows a fast anode discharge current ignition in order
to investigate the ion dynamics during forced and free low frequency current oscillations. The
current is switched oﬀ during 5 µs at a 2 kHz repetition rate without any synchronization with
the discharge current oscillation cycle. The outline of this paper is as follows. In Sec. 2, the
LIF optical assembly is described and the pulse-counting technique is introduced. Section 3
shows contour plots of the time-varying Xe+ion VDF as well as traces of various velocity
groups at the thruster outlet. In Sec.4, the temporal behavior of macroscopic quantities
like the density, the mean velocity and the velocity dispersion, is examined and discussed.
Section 5 reports on variation over time of the accelerating electric ﬁeld in the crossed-ﬁeld
discharge of a Hall thruster. Finally, concluding remarks will be presented in Sec. 6.
3
2 Diagnostic technique
2.1 Optical bench and collection branch
The LIF optical assembly is extensively described in Ref. [15]. The laser beam used to
excite Xe+metastable ions at 834.7 nm is produced by an ampliﬁed tunable single-mode
external cavity laser diode. The wavelength is accurately measured by means of a calibrated
wavemeter whose absolute accuracy is better than 100 MHz, which corresponds to 90 m/s.
A plane scanning Fabry-Prot interferometer with a 1.29 GHz free spectral range is used to
real-time check the quality of the laser mode and to detect mode hops. The primary laser
beam is modulated by a mechanical chopper at a low frequency 20 Hz before being coupled
into a 50 m long optical ﬁb er of 50 µm core diameter. The ﬁber output is located behind the
thruster. Collimation optics are used to form a narrow beam that passes through a small hole
located at the back of the PPSrX000 thruster. The laser beam propagates along the channel
axis in the direction of the ion ﬂow. Typically, the laser power density reaches 3mW/mm2,
which warrants a weak saturation eﬀect on the studied transition.
A collection branch made of a 40 mm focal length lens, which focuses the ﬂuorescence light
onto a 200 µm core diameter optical ﬁber, is mounted onto a travel stage perpendicular to
the channel axis. The magniﬁcation ratio is 1, meaning that the spatial resolution is 200µm
in axial direction. A 16 mm long slit was made in the channel dielectric outer wall in order to
carry out measurements inside the channel. The ﬂuorescence light transported by the 200 µm
ﬁber is focused onto the entrance slit of a 20 cm focal length monochromator that isolates the
541.9 nm line from the rest of the spectrum. A photomultiplier tube serves as a light detector.
2.2 Lock-in photon-counting device
The pulse (or photon) counting technique allows the detection of a very low level signal with
an excellent time resolution. When combined with a modulation of the laser light intensity,
the pulse counting technique can distinguish between LIF photons and spontaneous emission
photons. The technique is known as the time-resolved pulse counting lock-in detection tech-
nique. In this work, a customized pulse-counting system is used to measure time-dependent
ion VDF [16]. Here we brieﬂy outline the main characteristics and settings of the system.
A block diagram of the pulse counting system is shown in Fig. 2. Photons are detected by
means of a high gain and low dark noise PMT (R7518-P from Hamamatsu). A fast ampliﬁer
and discriminator module (9302 from Ortec - 100 MHz counting rate) is used to screen out
dark current from PMT dynodes, to limit the pulse rate thereby avoiding saturation of the
pulse counter, and to transform any single event – here the arrival of a photon – into either
a NIM or a TTL pulse. Pulses are subsequently treated by the lock-in pulse counter device,
which counts events as a function of time. A trigger starts the counter which segments photon
count data into sequential time bins. Notice that up to 32 kbins are available. The width of
the bins can be set from 10 ns to 655 s. The instrument records the number of photons that
arrive in each bin. In order to greatly improve the signal-to-noise ratio, the counter is able
to operate in real-time addition-subtraction mode. The laser beam intensity is modulated at
low frequency (20 Hz) by means of a mechanical chopper. Each pulse recorded when the
laser is propagating through the plasma (laser-on mode) is added to the time series; the signal
corresponds to LIF photons plus background photons. Each pulse recorded when the laser
is suspended (laser-oﬀ mode) is subtracted from the time series: in that case the signal is
solely composed of background photons. A 2 kHz trigger signal generated by the counter itself
4
was used to deﬁne the start of the measurement cycle. The time resolution, i.e. the width
of each bin, was set to 100 ns and 5000 bins were used. The duration of one measurement
cycle is therefore 500 µs, corresponding to about 6 times the period of low-frequency current
oscillations of the PPSrX000 thruster operating at 500 V and 6 mg/s. In order to obtain a
reasonable signal-to-noise ratio, light was accumulated over 1 million cycles.
The procedure to obtain the time-resolved ion VDF is the following:
the laser is ﬁxed at a given wavelength λcorresponding to a certain ion velocity group
δv. The extent (dispersion) of the velocity group results from the spectral width of the
laser beam and from the thermal expansion of the laser cavity. A feedback loop allows a
minimization of the shift of the laser wavelength. The dispersion is thus around 10 m/s.
The pulse counter is used to record the number of ﬂuorescence photons induced by
excitation of metastable ions at λ. That means we follow the temporal evolution of the
velocity group δv.
After about one million discharge current disconnection cycles the laser wavelength is
changed and a new measurement starts.
To obtain a smooth ion VDF about 15 to 20 diﬀerent wavelengths are used.
3 Temporal characteristics of the ion VDF
3.1 Experimental conditions - Ion emission light
All measurements were performed in the PIVOINE-2g test-bench. The PPSrX000 Hall eﬀect
thruster was equipped with BN-SiO2channel walls and with a carbon anode. All thruster
parameters were kept unchanged during the experiments: The applied voltage Udwas set to
500 V, the anode xenon mass ﬂow rate Φawas ﬁxed at 6 mg/s and the magnetic ﬁeld strength
Bwas 150 G. The mean discharge current is 5.4 A and the oscillation frequency is found to
be 13.7 kHz. The temporal characteristics of the ion VDF were recorded for three locations
along the channel centerline. Each position deﬁnes a distinct area in terms of electric ﬁeld
magnitude [15]. The position x= -15 mm corresponds to a zone through which the electric
ﬁeld is almost zero and the ionization is strong. At x= -2.5mm, the electric ﬁeld is large and
ionization is maintained. The channel exit plane, x= 0 mm, corresponds to a region where
ionization ceases and the electric ﬁeld strength is high with E300 V/cm.
In this study, the ion ﬂow dynamics was investigated before and after a fast shut-down of
the anode discharge current. It was therefore possible to examine the temporal characteristics
of the ion VDF during plasma breakdown and ignition as well as during forced and free current
oscillations. The anode current is switched oﬀ during 5 µs at 2 kHz by way of an optically
controlled fast power switch based on a MOSFET [9]. The power switch is directly driven by
the counter, as shown in Fig. 2. There is no synchronization between the power cut cycle and
the discharge current waveform. In other words, the anode current is switched oﬀ randomly
at any time with respect to the discharge current natural oscillations.
Figure 3 displays the intensity of the 541.9 nm ion line as a function of time observed
at x= -2.5 mm recorded with the photon-counting device operating in addition mode only.
A snapshot of the anode current waveform is also shown. Measurement reveals the time
5
evolution of natural plasma emission. The power is always switched oﬀ at t= 0 µs (reference
time). The plasma decays with a 1/etime of 2.1 µs. However, it does not fully vanish as
charged-particle recombination and diﬀusion processes timescales are not inﬁnitely shorter
compared to the 5 µs current-oﬀ time period. On the contrary, both the anode current and
the Hall current cancel in a few hundreds of ns [17]. This point is speciﬁcally addressed
in the next sections. A large amount of light is quickly produced at the re-ignition stage,
as can be seen in Fig. 3. In like manner, a discharge current burst always occurs at re-
ignition [9, 17]. This forced plasma oscillation originates in the sudden ionization of the great
amount of propellant atoms accumulated inside the channel when the discharge is oﬀ. The
plasma oscillates with a mean period of about 83 µs that corresponds to a 12 kHz frequency.
As can be observed in Fig. 3, the amplitude of light oscillations diminishes with time, and
the signal ﬁnally approaches a constant level. This phenomenon is due to the fact that, the
current disconnection cycle is not synchronized with the natural current oscillations, while
data acquisition is a cumulative process over thousand of cycles.
3.2 Contour map
Figure 4 shows contour plots of the time evolution of the Xe+ion velocity distribution func-
tions for three positions along the thruster channel axis, respectively x= -15 mm, x= -2.5 mm,
and x= 0 mm. All velocity groups vanish quickly after the current is switched oﬀ on a time
scale on the order of a few 100 ns. This property indicates the electric ﬁeld cancels almost
instantaneously when the current is stopped. As a consequence, the discharge current as
well as the Hall current disappear over an extremely short time period, as experimentally
observed [17]. Yet, very slow ions do not fully disapp ear in 5 µs as recombination, ambipolar
diﬀusion to walls and drift out of the acceleration zone are slow processes. In Fig. 4, the
ﬂuorescence signal is signiﬁcantly above zero after 5 µs for x= -15 mm. The remark holds
also true for the ion emission signal, see Fig. 3. Note that a Xe+ion travels 1.5 mm in 5 µs
at the thermal speed vth 300 m/s. Very slow ions, i.e ions with the atom speed, are always
produced ﬁrst at re-ignition as electrostatic acceleration is not an instantaneous process. In
Fig. 4, slowest ions are indeed observed ﬁrst and the mean velocity gradually increases up
to a limit. Ions moving with a velocity close to the thermal speed are not visible in Fig. 4
for x= -2.5 mm and x= 0 mm as the ion VDF is truncated due to a lack of data points.
The large production of ions at plasma ignition originates from the fact that the channel
is entirely ﬁlled up with xenon atoms, as previously explained. As can be seen in Fig. 4,
the Xe+ion VDF in axial direction changes in time, especially during the ﬁrst two plasma
oscillations. This phenomenon indicates that the acceleration potential, and thus the electric
ﬁeld, is likely to vary in time with a frequency on the order of the main discharge current
oscillation frequency.
3.3 Velocity groups
Examination of the temporal characteristics of the ion VDFs provides a general outline of
the ion dynamics in the discharge of a Hall thruster. In contrast, a critical analysis of the
temporal behavior of ion velocity groups reveals in great detail the intricate character of the
physics at work. The time evolution at the channel outlet of eight well-identiﬁed ionic velocity
groups δv is shown in Fig. 5. The temporal evolution is monitored at the thruster channel
exhaust. Graphs correspond to horizontal cross-sections of the lower contour plots in Fig. 4.
6
All velocity groups quickly vanish as soon as the power is switched-oﬀ: the mean 1/edecay
time for all δv is 1.5 µs. The discharge current as well as the Hall current decay in about
the same time period [17]. Nonetheless, fastest ions disappear out of the acceleration region
ﬁrst: the 1/edecay time is 1.1 µs at 12960 m/s. On the contrary, slowest ions are produced
ﬁrst when the discharge current is re-ignited. In Fig. 5, the group δv = 9515 m/s is detected
at 9 µs and it reaches its highest amplitude at 19 µs whereas the group δv = 13390 m/s is
detected at 20 µs and it goes through a maximum at 39 µs. All velocity families oscillate
nonetheless with the same period of time T73 µs, that means with a frequency around
14 kHz.
When the power is turned on again, ionization immediately takes place, see Fig. 3. Dis-
charge and Hall current are restored in less than 1 µs and ions are gradually accelerated as
shown in Fig. 5. All these facts indicate that the electric ﬁeld is established on a microsecond
timescale at re-ignition. However, it seems at ﬁrst sight in manifest contradiction with the
time it takes to detect fast ions inside the thruster channel. The time of ﬂight of a given
ion velocity group across the acceleration layer after plasma re-ignition can be assessed by
numerically solving the particle motion equation in an external electric ﬁeld E:
(1) dv =e
mE dt,
where eis the elementary charge and mis the mass of a xenon ion. For simplicity’s sake,
the electric ﬁeld is ﬁrst set constant in space and in time through the whole acceleration
region (from x= -10 mm to 20 mm) with a magnitude of 170 V/cm [15]. Moreover, the
ﬁeld is assumed to be instantaneously created and the ionization process is stationary and
homogeneous. The initial velocity v0is ﬁxed as the thermal speed vth. Computations reveal
that the group δv = 13390 m/s appears at the exit plane at t= 1.05 µs. Ions have then
travelled 7.2 mm, a distance compatible with the acceleration layer size L30 mm. However,
this velocity group is observed ﬁrst at t= 20 µs, see Fig. 5. Another approach consists of
taking a steady electric ﬁeld distribution similar to the measured one [15]. When ions are
created about 19 mm inside the channel, computations indicate that the group δv = 13390 m/s
is indeed observed at x= 0 mm for t20 µs. Yet ions moving at 9515 m/s are solely seen a
few hundreds of ns earlier, in contradiction with experimental outcomes.
One way to better duplicate reality consists for instance of considering that the ionization
proﬁle or the electric ﬁeld can evolve in the course of time. With a stationary electric ﬁeld
proﬁle similar to the measured one [15], numerical simulations using Eq. 1 show that ions
must be created ﬁrst in the vicinity of the channel outlet and the ionization front must move
towards the anode with a speed on the order of 500 m/s to reproduce trends that are ex-
perimentally observed in Fig. 5. Propagation of an ionization wave through the acceleration
layer was proposed to explain experimental results acquired by means of time-resolved optical
emission spectroscopy with a ﬁber comb [11] as well as CCD images of the plume behavior
with a sp eed around 1-2 km/s.
Figure 6 shows the time evolution of the velocity group δv = 9550 m/s for two locations,
respectively x= -2.5 mm and x= 0 mm. The ion family is ﬁrst observed at the channel
exit plane at t= 8 µs before being detected inside the channel at t25 µs. The highest
amplitude is attained for t= 19 µs and t= 45 µs at x= 0 mm and x= -2.5 mm, respectively.
Results indicate that ions are created within a broad region inside the channel: ions created
close to the exit plane are indeed detected ﬁrst. Nevertheless, experimental outcomes cannot
7
be correctly simulated when assuming that an ionization wave travels from the plume near
ﬁeld towards the anode while the electric ﬁeld distribution stays unchanged. On the contrary,
results suggest that the ionization front as well as the acceleration layer change in time to-
gether. Numerical simulations, though basic, and experimental data are therefore in favor of
complex plasma dynamics within the discharge of a Hall thruster. It is worth noticing that
the behavior of Xe+ion velocity groups during a forced oscillation that follows a power-oﬀ
period may not directly image the normal behavior during a natural plasma oscillation from
the viewpoint of amplitude of the observed phenomena.
4 Evolution in time of the density, mean velocity and disper-
sion
The time evolution of various averaged quantities is plotted in Fig. 7, 8 and 9 for three
positions along the channel axis, respectively x=15 mm, x=2.5 mm and x= 0 mm.
The relative metastable Xe+ion density is given by the area of the VDF. The mean velocity
and the velocity dispersion are computed from, respectively, the ﬁrst and the second order
moments of the velocity distribution. The velocity spread is in fact expressed in terms of p
parameter [14, 15]. The latter reads:
(2) p= 2 p2Ln(2) ×σ2.335 ×σ,
where σis the standard deviation. The quantity pis equal to the FWHM in the case of a
Gaussian proﬁle.
As can be seen in Fig. 7, the ion density is oscillating in time with a frequency of about
16 kHz whatever the position. Changes in the ion density are connected with temporal charac-
teristics of the ionization rate. The latter is driven by a prey-predator kind of process between
atoms and charged particles [6, 18, 19]. Besides, Langmuir probe measurements have shown
that the electron density periodically varies in time with the discharge current oscillation
frequency [10]. The fact that in Fig. 7, the ﬁrst maximum of the ion density waveform is
shifted in time with respect to the ﬁrst maximum of the 541.9 nm line intensity proﬁle for
x=2.5 mm and x= 0 mm is an experimental artefact due to the truncation of the measured
VDF.
The time evolution of the mean velocity is shown in Fig. 8. At x= -15 mm, the velocity
slightly oscillates around zero as the observation point is outside the acceleration layer. At
x=2.5 mm and x= 0 mm, the mean velocity oscillates around a constant value, which
corresponds with a reasonably good agreement to the value obtained by means of time-
averaged LIF spectroscopy at the same location [15]. The small diﬀerence is due to the
power-switch disturbance. As shown in Fig. 8, the velocity changes in time are however
small. In terms of kinetic energy the largest gap is 10 eV at -2.5 mm, respectively 25 eV at
the channel exit plane. Energy variation of a few tens of eV during a low-frequency current
oscillation cycle was recorded in the plume far-ﬁeld of a SPT100-ML Hall thruster using
a repulsing potential analyzer [11]. Hybrid ﬂuid/kinetic models also predict low-amplitude
periodical variation of the ion velocity [18, 19] with a strong correlation between discharge
current and ion velocity temporal behavior. The observed oscillations of the mean ion axial
velocity can solely be explained by a back and forth motion of the ionization and acceleration
layers and/or a change in the electric ﬁeld magnitude, as already suggested by the analysis
of ion velocity groups characteristics as a function of time.
8
As can be seen in Fig. 9, the velocity dispersion does not vary much in time other than
at current shut oﬀ and restart. At the thruster channel exhaust, the velocity dispersion (p
parameter) is almost constant with a value of 2950 m/s. In terms of energy spread it is 47 eV.
In previous investigations [14, 15], it was clearly demonstrated that in a Hall thruster environ-
ment the velocity dispersion originates mostly in the spatial overlap between the ionization
and the acceleration layers. As the velocity dispersion stays unchanged with time whereas
the ion density and velocity do vary, one can conclude the ionization front and the electric
ﬁeld distribution have a correlated dynamics. For instance, assuming the form of the two
proﬁles stay unchanged, the two layers would have to move along at once together in the
axial direction. Actually, computer simulations indicate that both the shape and the location
of the ionization and electric ﬁeld proﬁles change with time [18].
5 Low-frequency electric ﬁeld oscillations
The oscillation in time of the accelerating electric ﬁeld can be assessed from the time-
dependent proﬁle of the Xe+ion mean velocity ¯v, see Fig. 8, assuming a collisionless medium.
The electric ﬁeld is computed according to the formula:
(3) E(t) = mXe+
2e¯v(x1, t)2¯v(x2, t)2
dx1x2,
where xis the position and drefers to the distance. The time-dependent electric ﬁeld is plotted
in Fig. 10 for the area that ranges between -2.5 mm and the exit plane. The electric ﬁeld is
found to oscillate with a period T90 µs, i.e. f11 kHz, around a mean value of about
215 V/cm disregarding the power-oﬀ period. This value is close to the value of 245 V/cm
found by way of time-averaged laser spectroscopy [15]. The amplitude of the ﬁeld oscillations
is, however, relatively weak. Over the ﬁrst free oscillation that extends from t= 75 µs until
t= 165 µs the amplitude varies at most from 190 V/cm to 240 V/cm, therefore, the electric
ﬁeld variation is in the range ±10 % ahead of the thruster channel exhaust. Finally, in Fig. 10,
one can also notice the slow rise of the electric ﬁeld magnitude from 100 V/cm at 10 µs to
250 V/cm at 30 µs. This temporal evolution is likely to be connected with the time it takes
for the establishment of an equilibrium state for both the discharge and the Hall currents.
6 Conclusions
The examination of the temporal characteristics of the Xe+ion velocity distribution func-
tion in the magnetized discharge of a Hall eﬀect thruster reveal the complex dynamics of
the ionization and the acceleration processes. Three results are especially of great interest.
First, the ionization proﬁle and the electric ﬁeld distribution are unstationary. Second, the
ionization and acceleration layers vary in time in such a way that their spatial overlap stays
almost unchanged as the ion velocity spread does not change much after ignition. Third, the
amplitude of the low-frequency electric ﬁeld oscillations ahead of the channel exit plane is
weak.
Even though this study has brought new facts about the physics of a Hall eﬀect thruster,
it appears necessary to carry on this type of experiments aiming at building-up of a larger
set of data. The latter is actually necessary to improve our understanding of the time and
space evolution of the electric ﬁeld in a Hall thruster. Measurements of the time-dependent
9
ion VDF must be performed with a better signal-to-noise ratio at many locations along
the thruster channel centerline. Measurements must also be carried out for several thruster
operating conditions and various power levels. Finally, experimental results must be critically
compared with numerical outcomes of PIC models of Hall thruster behavior.
Acknowledgements
Works are performed in the frame of the joint-program CNRS/CNES/SNECMA/Universities
3161 entitled “Propulsion par plasma dans l’espace”. They are also ﬁnancially supported by
the French National Research Agency in the frame of the 06-BLAN-0171 TELIOPEH project.
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11
Figure 1: Cross-section view of a Hall eﬀect thruster. The symbol estands for electron, afor atom
and ifor ion. The channel exit plane is referred to as x= 0 in this work.
12
Figure 2: Block diagram of the lock-in pulse counting system used in this work to measure the time-
resolved Xe+ion VDF by means of LIF spectroscopy. The PMT is placed behind a 20 cm focal length
monochromator. The anode discharge current switch is externally driven by the counter.
13
Figure 3: (Top) Snapshot of the anode current waveform. The discharge current is switched for 5 µs
at t= 0 µs. (Bottom) Change in time of the ion emission at 541.9 nm observed using the photon
counting technique at x= -2.5 mm. The inset panel displays enlargement of the emission signal from
50 µs until 350 µs. The oscillation frequency is 12 kHz.
14
Figure 4: Contour map of the Xe+ion axial VDF as a function of time for three locations along the
channel axis of the PPSrX000 Hall thruster ﬁred at 500 V: x= -15 mm, x= -2.5 mm and x= 0 mm.
15
Figure 5: Trace of the time evolution of eight ionic velocity groups at the channel exit plane (x= 0
mm).
16
Figure 6: Time evolution of the velocity group δv 9 550 m/s for two lo cations: x=-2.5 mm and
δv = 9605 m/s (solid line) and x= 0 mm and δv= 9515 m/s (circle).
17
Figure 7: Time evolution of the metastable Xe+ion relative density for three positions along the
channel axis of the PPSrX000 Hall thruster. The density is given by the VDF area. Also shown is
the emission proﬁle at 541.9 nm (blue line).
18
Figure 8: Time evolution of the Xe+ion mean axial velocity for three on-axis positions. The time-
averaged velocity measured in a previous study at x=2.5 mm and 0 mm is 9300 m/s and 13100 m/s,
respectively [15]. Also shown is the proﬁle of the light emission at 541.9 nm (blue line).
19
Figure 9: Time evolution of the Xe+ion axial velocity dispersion (pparameter) for three positions
along the channel axis of the PPSrX000 Hall thruster. Also shown is the proﬁle of the light emission
at 541.9 nm (blue line).
20
Figure 10: Electric ﬁeld temporal characteristics ahead of the PPSrX000 thruster channel exhaust
(between 0 and -2.5 mm). The ﬁeld strength is determined from the mean Xe+ion axial velocity.
21
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Closed drift thrusters are reviewed. The publications on these thrusters constitute a large body of information. This article can therefore include only the most prominent theoretical and experimental features of closed drift thrusters. In some regards, this article is also an attempted synthesis of the differing views of these thrusters found in literature, as well as in our own work. In a closed drift thruster, the electric field that accelerates the ions is established by an electron current that passes through and is impeded by a magnetic field. The precessing electrons in this magnetic field follow a closed drift path giving this thruster its name. Closed drift thrusters are divided into magnetic layer and anode layer types, based both on the geometrical and material differences in the discharge channels of the two types, and on the different physical processes that take place within the discharge plasma. Considered as a whole, the publications on closed drift thrusters constitute an impressive body of information that, for the most part, was generated in Russia independently of US research on electric thrusters.
Article
Full-text available
Hall-effect thruster plasma oscillations recorded by means of probes located at the channel exit are analyzed using the empirical mode decomposition EMD method. This self-adaptive technique permits to decompose a nonstationary signal into a set of intrinsic modes, and acts as a very efficient filter allowing to separate contributions of different underlying physical mechanisms. Applying the Hilbert transform to the whole set of modes allows to identify peculiar events and to assign them a range of instantaneous frequency and power. In addition to 25 kHz breathing-type oscillations which are unambiguously identified, the EMD approach confirms the existence of oscillations with instantaneous frequencies in the range of 100– 500 kHz typical for ion transit-time oscillations. Modeling of high-frequency modes 10 MHz resulting from EMD of measured wave forms supports the idea that high-frequency plasma oscillations originate from electron-density perturbations propagating azimuthally with the electron drift velocity. © 2005 American Institute of Physics.
Article
Full-text available
The results of a study of laser-induced fluorescence velocimetry of neutral and singly ionized xenon in the plume and interior portions of the acceleration channel of a Hall thruster plasma discharge operating at powers ranging from 250 to 725W are described. Axial ion and neutral velocity profiles for four discharge voltage conditions (100V, 160V, 200V, 250V) are measured as are radial ion velocity profiles in the near-field plume. Ion velocity measurements of axial velocity both inside and outside the thruster as well as radial velocity measurements outside the thruster are performed using laser-induced fluorescence with nonresonant signal detection on the xenon ion 5d[4]7/2–6p[3]5/2 excitation transition while monitoring the signal from the 6s[2]3/2–6p[3]5/2transition. Neutral axial velocity measurements are similarly performed in the interior of the Hall thruster using the 6s[3/2]0 2–6p[3/2]2transition with resonance fluorescence collection. Optical access to the interior of the Hall thruster is provided by a 1-mm-wide axial slot in the insulator outer wall. While the majority of the ion velocity measurements used partially saturated fluorescence to improve the signal-to-noise ratio, one radial trace of the ion transition was taken in the linear fluorescence region and yields a xenon ion translational temperature between 400 and 800K at a location 13mm into the plume.
Article
Time-resolved electrostatic probe measurements were performed in the near field of a SPT100-ML Hall effect thruster in order to investigate electron properties changes on a microsecond time scale. Such measurements allow one to monitor the electron temperature Te, the electron density ne, as well as the plasma potential Vp during a time period that corresponds to one cycle of a breathing-type plasma oscillation with f ≈ 15–30 kHz. Although Te(t) stays constant in time, ne(t) and Vp(t) oscillate with the discharge current waveform frequency. The observed time delay between ne and anode discharge current (Ida) waveforms, which is of approximately 7 μs, is linked to the ion transit time from the ionization layer to the probed near-field region. The same time gap is measured between Vp(t) and Ida(t), however Vp(t) and ne(t) are in phase opposition. The electron density reaches its highest value at the very moment ions are ejected out of the thruster discharge chamber, which also corresponds to the instant the cathode potential is the most negative. Such a behavior images the need for ion beam neutralization. Further, it is shown that there is a strong correlation between the electron dynamics and the presence of high frequency (HF) plasma oscillations in the megahertz range: HF fluctuations are the strongest when ne is the highest.
Article
Stationary plasma thrusters are ion thrusters whose properties make them especially suitable for satellite station keeping or orbit transfer. In these thrusters, a magnetic field transverse to the electron flow towards the anode increases the electron collision frequency and makes possible the generation of a plasma at relatively low gas flow and gas density. The decrease of the plasma conductivity due to the magnetic field induces a large electric field in the plasma which accelerates the quasicollisionless ions whose trajectories are not significantly affected by the magnetic field. The purpose of this article is to clarify, using results from a simple model, the electrical properties of these thrusters and the low frequency oscillation regime which has been observed experimentally. The model is based on the assumption of quasineutrality of the plasma column and on a 1D transient hybrid treatment of electron and ion transport in the device. Electrons are considered as a fluid and ions are described with a collisionless kinetic equation. This model provides reasonable estimates of the plasma properties and is able to give a clear picture of the low frequency oscillations, qualitatively close to the experimental observations. © 1998 American Institute of Physics.
Article
A discussion is presented on the results and predictive capabilities of a two-dimensional (2D) hybrid Hall effect thruster (HET) model. It is well known that classical (collision-induced) cross-field electron transport and energy losses are not sufficient to explain the observed HET characteristics. The 2D, quasineutral, hybrid discharge model uses empirical parameters to describe additional, anomalous electron transport and energy loss phenomena. It is shown that, for properly adjusted empirical parameters, the model can qualitatively reproduce the observed thruster behavior over a large range of operating conditions. The ionization and transit-time oscillations predicted by the model are described, and their consequences on the time-averaged thruster properties are discussed. Finally, the influence of the empirical parameters on the model results is shown, especially on quantities that can be measured experimentally. © 2004 American Institute of Physics.
Article
This paper presents some aspects of the research developed in the frame of a coordinated program launched in France in 1996 and devoted to plasma thrusters for space technologies. Relevant results of physical studies have been selected from the literature with the addition of recent original results. The thrusters within the scope of this research are diagnostic equipped versions of industrial realizations, in a thrust level range of 0.1 N and electrical power 1.5 kW. The optical and electrical diagnostics concern studies of the thruster plasma and of the thruster plume. Transient phenomena in these two regions, related to discharge current fluctuations or oscillations on a typical time scale of 40 µs, have been space-time characterized. This has been achieved by developing a large panel of diagnostics including RFEA, Langmuir probes, OES, fast camera imaging and electron drift Hall current probe. They lead to a coherent representation of these phenomena , in rather good qualitative agreement with 1D modelling. But they emphasize also the importance of 2D effects. Insights obtained through combined LIF (on Xe+ ions) and OES diagnostics are also presented. They concern the ionization-acceleration region in the thruster plasma, where intrusive diagnostics are disturbing in nature, and open a new step for a significant improvement of the detailed understanding of these thrusters. Such improvements are required when looking at the final goal of a predicable modelling simulation able to help the design of optimized structures at various thrust levels, in spite of the important work devoted to these devices in the former USSR and by Russian teams in Moscow at the MIREA, MAI-RIAME and KOURCHATOV Institutes.
Article
We describe a time‐resolved pulse‐counting system well adapted for the detection of continuous laser induced fluorescence (LIF) signals in repetitive phenomena, when a strong background emission is present. It consists of 256 channels coupled to a first in first out memory and interfaced to a 486 DX 33 PC, for data storage. It accepts time‐averaged count rates up to 450 kcount/s. Time between channels can be set from 12.5 ns to several μs and the dead time between two consecutive cycles of the physical phenomena is less than 20 ns. In phase with a chopper, which modulates the laser beam, it adds the observed photon signal to the channel memories when the beam is on and substracts it when the beam is stopped, acting like a lock‐in amplifier which detect only the modulated part of the signal. The minimum detectivity on the LIF signal is only limited by the shot noise of the plasma induced emission signal. As an application, we studied the time variation of the Ar<sup>+</sup>*(<sup>2</sup>G 9/2 ) metastable ions, detected by LIF, in two types of plasmas. Their radiative lifetime and collisional quenching frequencies were deduced from their decay rate in the afterglow of a pulsed Helicon reactor. We also observed the evolution of their density in a 455 kHz capacitively coupled argon discharge. © 1996 American Institute of Physics.

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text:
Remark 50.23.1. Let $X \to S$ be a morphism of schemes. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Let $Z \subset X$ be the closed subscheme cut out by $f_1, \ldots , f_ c$. Below we will study the gysin map

50.23.1.1
$$\label{derham-equation-gysin} \gamma ^ p_{f_1, \ldots , f_ c} : \Omega ^ p_{Z/S} \longrightarrow \mathcal{H}_ Z^ c(\Omega ^{p + c}_{X/S})$$

defined as follows. Given a local section $\omega$ of $\Omega ^ p_{Z/S}$ which is the restriction of a section $\tilde\omega$ of $\Omega ^ p_{X/S}$ we set

$\gamma ^ p_{f_1, \ldots , f_ c}(\omega ) = c_{f_1, \ldots , f_ c}(\tilde\omega |_ Z) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$

where $c_{f_1, \ldots , f_ c} : \Omega ^ p_{X/S} \otimes \mathcal{O}_ Z \to \mathcal{H}_ Z^ c(\Omega ^ p_{X/S})$ is the map constructed in Derived Categories of Schemes, Remark 36.6.10. This is well defined: given $\omega$ we can change our choice of $\tilde\omega$ by elements of the form $\sum f_ i \omega '_ i + \sum \text{d}(f_ i) \wedge \omega ''_ i$ which are mapped to zero by the construction.

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# Sum of a subsequence of a sequence with infinite sum

Let $(b_n)$ be a sequence where $\forall n \in \mathbb N \ 0 < b_{n+1} < b_n$ and $b_n \rightarrow 0$ as $n \rightarrow \infty$. Furthermore suppose that $\sum \limits_{n=1}^\infty b_n = \infty$.

What can I say about $\sum \limits_{n=9k, \ k \in \mathbb N}^\infty b_n$?

I'll refer to $B_m = \sum \limits_{n=1}^m b_n$ and $S_m = \sum \limits_{n=9k, \ k \in \mathbb N}^m b_n$.

I know that a monotonic and divergent sequence has no convergent subsequence. But my problem here is that $S_m$ is not a subsequence of $B_m$ so I don't know what tools to apply.

Examples I've been considering are the harmonic series and $b_n = \log{\frac{n}{n-1}}$.

This isn't homework, it's just related to something that I've been wondering about. Thanks for any help.

• The sub-sum diverges. Since your sequence is decreasing and positive, you can bound your original sum from above and below by suitable multiples of your sub-sum (plus some additive constant). So the sub-sum and the original sum either converges at the same time or diverges at the same time. – achille hui Sep 16 '15 at 14:53

Let $b_0 := b_1 +1$ (this makes notation a little more easy, but does not change the argument). Note that, as $(b_n)$ is decreasing, that \begin{align*} \sum_{n=0}^\infty b_n &= \sum_{k=0}^\infty \sum_{i=0}^8 b_{9k+i} \\ &\le \sum_{k=0}^\infty \sum_{i=0}^8 b_{9k}\\ &= 9\sum_{k=0}^\infty b_{9k} \end{align*} Hence, as $\sum_{n=0}^\infty b_n = \infty$, we have $\sum_{k=0}^\infty b_{9k} = \infty$.


text:
## Algebra 1

Published by Prentice Hall

# Chapter 8 - Polynomials and Factoring - 8-6 Factoring ax(squared) + bx + c - Mixed Review - Page 510: 61

d=12

#### Work Step by Step

In order to solve for d the proportion $\frac{2}{3}$=$\frac{d}{18}$, we must apply the Cross Products Property of a Proportion that states that if $\frac{a}{b}$=$\frac{c}{d}$, where b$\ne$0 and d$\ne$0, then ad=bc. We'll set a=2, b=3, c=d and d=18, and apply the Property. $\frac{5}{6}$=$\frac{x}{15}$ (2)(18)=(d)(3) 36=3d Then, to solve for d, we'll divide by 3 on both sides of the equation $\frac{36}{3}$=$\frac{3d}{3}$ d=$\frac{36}{3}$=12

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Quality Assurance and Energy Calibration of Calorimeters
Cocktail Running and External Input
Neutral Meson and Direct Photon Analysis - Afterburners
General Naming Scheme and Analysis Tasks
In our framework we follow a common naming scheme and several abbreviations will occur frequently. When we are talking about the meson analysis, we usually refer to the neutal pion and eta analysis. While when we are talking about the photon analysis we generally mean the direct photon analysis. Within the software package different photon reconstruction techniques are implemented:
Abbreviation
Explanation
Conversions (Conv, PCM)
Photon reconstruction via conversion in the detector material based on the electrons and positrons reconstructed in the TPC
Calo (EMCal, DCal PHOS)
Photons reconstructed via their energy deposit in the calorimeters
These are then combined to reconstruct the neutral mesons in either the two photon (
$\gamma \gamma$
) or the Dalitz decay channel (
$\gamma e^+ e^-$
) using the same or different techniques or detectors for the photon reconstruction. The corresponding analysis tasks in AliPhysics in PWGGA/GammaConv are:
• AliAnalysisTaskGammaConvV1.cxx
• AliAnalysisTaskGammaCalo.cxx
• AliAnalysisTaskGammaConvCalo.cxx
• AliAnalysisTaskGammaConvDalitzV1.cxx
• AliAnalysisTaskGammaCaloDalitzV1.cxx
• AliAnalysisTaskGammaCaloMerged.cxx
These tasks are named according to their photon/meson reconstruction method and will in most cases be used to reconstruct not only the
$\pi^0$
and
$\eta$
meson but also the direct photons. A similar naming scheme will be used throughout the whole software package and is explained in the following.
Abbreviation
Explanation
Conv, PCM
Photons reconstructed using conversions and then paired with other photons reconstructed with the same technique
Calo
Photons reconstructed using one of the three calorimeters (EMCal, DCal, PHOS) and then paired with other photons reconstructed with the same technique
ConvCalo
Photons reconstructed using one of the three calorimeters (EMCal, DCal, PHOS) and then paired with photons, which have been reconstructed from conversions. Currently 3 combinations are possible: PCM-EMC, PCM-DMC, PCM-PHOS. For the direct photon reconstruction the conversion photon is used for the inclusive photon reconstruction.
ConvDalitz
Photons reconstructed using conversions and then paired with 2 primary electrons reconstructed in the TPC and ITS.
CaloDalitz
Photons reconstructed using one of the three calorimeters (EMCal, DCal, PHOS) and then paired with 2 primary electrons reconstructed in the TPC and ITS.
CaloMerged, Merged
Neutral pions and eta mesons are reconstructed in the calorimeter via single clusters in the region where the 2 photons can no longer be separated due to the calorimeter resolution. These techniques are also refered to as mEMC, mDMC or mPHOS, where only the first has been explored so far.
They are configure by the AddTasks in PWGGA/GammaConv/macros following a similar naming scheme:
AddTask_Gamma[Calo,ConvV1,ConvCalo,ConvDalitzV1,CaloDalitzV1,CaloMerged]_[pp,pPb,PbPb].C
These tasks produce output files which are named according to the reconstruction method and the train-configuration which has been chosen i.e.: _GammaConvV1_$TRAINCONFIG.root, GammaCalo_$TRAINCONFIG.root, GammaConvCalo_$TRAINCONFIG.root, GammaCaloDalitz_$TRAINCONFIG.root, GammaConvDalitzV1_$TRAINCONFIG.root, GammaCaloMerged_$TRAINCONFIG.root_
For the heavier meson reconstruction using the
$\pi^0$
in the decay chain the same convention for the names could not be kept and the corresponding tasks are named according to the decay chain and meson which they are supposed to reconstruct:
• AliAnalysisTaskK0toPi0Pi0.cxx
• AliAnalysisTaskOmegaToPiZeroGamma.cxx
• AliAnalysisTaskNeutralMesonToPiPlPiMiPiZero.cxx
• AliAnalysisTaskEtaToPiPlPiMiGamma.cxx
In these cases the photon/neutral pion reconstruction technique is selected through the configurations in the AddTask, which follow a similar naming scheme as explained before. There are several "helper-tasks" implemented in PWGGA/GammaConv in addition, which are usually used for QA purposes, Cocktail or MC investigations. Furthermore they can be used to determine the material budget uncertainty.
• AliAnalysisTaskConversionQA.cxx
• AliAnalysisTaskGammaCocktailMC.cxx
• AliAnalysisTaskHadronicCocktailMC.cxx
• AliAnalysisTaskGammaPureMC.cxx
• AliAnalysisTaskResolution.cxx
• AliAnalysisTaskMaterial.cxx
• AliAnalysisTaskMaterialHistos.cxx
For the conversion analysis also the
$v_n$
reconstruction for the photons and pions is implemented in PWGGA/GammaConv, which is not yet the case for the corresponding calorimeter based analysis.
• AliAnalysisTaskGammaConvFlow.cxx
• AliAnalysisTaskPi0v2.cxx
Last modified 8mo ago
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# Tag Info

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### Why is padding used in CBC mode?

So, I'll answer the theoretical part of your question, since we need a key to address the practical part. Why is padding used in CBC? Blockcipher such as AES are encrypting blocks of a fixed given ...
• 7,266
Accepted

### Why can't we use the first block of AES-CBC as MAC

The MAC value should be calculated over all of the input, not just the first block. The chaining of CBC makes sure that the bits in the last block of ciphertext depends on all the previous blocks.
• 85.8k
Accepted

### What is the correct way to implement PBKDF2 + AES CBC + HMAC?

You'll find there's a lot of splitting hairs regarding this topic, especially key derivation. But yes, your pseudocode is fine, although you may want to revise (0, 128) => (0, 127) and (129, 256) => (...
• 3,695
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### Insecurity of CBC-MAC

Can anyone explain why CBC-MAC is not secure for variable length message? For the previous question I'll quote Matthew Green's post from 2013: A quick reminder. CBC-MAC is very similar to the ...
• 3,225
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### MAC using a modified CBC mode of operation

This scheme is not worth the name MAC; it is horribly weak. First and foremost, the tag/MAC is unchanged when two blocks of plaintext are exchanged (because of the commutativity and associativity of ...
• 125k
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### When do I need to use CBC and HMAC?

Also, there is CBC-MAC for providing integrity and confidentiality. Which one is better CBC-MAC or CBC with HMAC? Generally, asking which one is better results in opinionated answers. However, since ...
• 85.8k
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### CBC-MAC just to verify integrity

Yes, this is exactly what a message authentication code is for. Its job is to prevent an attacker from tampering with your message, or from forging completely bogus messages. For a secure MAC, it ...
• 44.5k
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### Are TLS 1.2 AES-CBC and AES-GCM Authenticated Encryption ciphersuites?

When can I consider a ciphersuite an Authenticated Encryption? To cite from Wikipedia: Authenticated Encryption: Authenticated encryption ... is a form of encryption which simultaneously provides ...
• 1,298
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### Why does the CBC-MAC require PRFs?

The question as posed (in the book) is a bit weird, mainly because it does not state that $F$ is required to be length preserving, however for the CBC-MAC construction to make sense it clearly has to ...
• 6,424

### CBC-MAC based on PRF

The CBC-MAC construction indeed can use a PRF instead of PRP. It is now based on PRP due to historical reasons: the blockciphers used for CBC-MAC were based on permutations. From the security point ...
• 5,547
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### How can CBC-MAC be secure when message length is fixed?

What you think of is called an extension attack and it turns out that this is the way to go if you would like to break the general CBC-MAC when the message length is not fixed. All that an adversary ...
• 181

### What is the correct way to implement PBKDF2 + AES CBC + HMAC?

IMO, you code looks pretty solid. A few things I might suggest taking a closer look at are: You haven't specified what iteration count you're using for PBKDF2. You should make the iteration count ...
• 44.5k

### Having CBC-MAC of a one block message, forge CBC-MAC of two blocks

Given just $a$ and $M(a)$, you can construct the message $a\,\|\,b$, where $b = a \oplus M(a)$, which gives the same CBC-MAC output $M(a\,\|\,b) = M(a)$ as $a$ itself. Indeed, we can append the block ...
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### Why are there no MACs inspired by block cipher modes other than CBC and CFB?

It is certainly wrong to state that "MAC can only be produced with AES in CBC and CFB mode", but there seems to be a simple reason that people were inspired by these modes when thinking up possible ...
• 11.2k

### Real time video stream AES encryption with authentication

updated per comments; Currently Netflix Uses AES-GCM I am now studying the AES encryption for real-time video stream. It seems that Netflix uses the AES-GCM (or CBC + MAC) mode for real-time video ...
• 43.5k
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### Security of a simple AEAD construction

Yes, this is secure. (one of the few cases where I'm pretty confident about this). Here are the arguments: Combining a secure (e.g. SUF-CMA) MAC with a secure (e.g. CPA-secure) encryption method in ...
• 44.7k
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### Is XCBC where k2 and k3 might be the identical, secure?

If key 2 and key 3 has a nonnegligible chance to be the same, then the attacker has a nonnegligible chance of being able to generate a valid (Message, MAC) pair. Here's how it works, if the message ...
• 133k


text:
# Using Matlab to plot a phase potrait for ODEs

#### dwsmith

##### Well-known member
First create the function file and name it whatever you would like. I prefer phase-portrait.

Code:
% Phase Plot Program
% To use this function, do the following:
% >> phase_portrait(x1, x2, y1, y2, tfinal, 'F', N);     for example,
% >> phase_portrait(-5, 5, -5, 5, 10, 'F', 5)

function [] = phase(x1, x2, y1, y2, tfinal, F, N);

%   x1 is the x-min value
%   x2 is the x-max value
%   y1 is the y-min value
%   y2 is the y-max value
%   tfinal is the length of time interval
%   F is the system function input as a string 'F'
%   NxN: is the number of orbits plotted

figure; hold on;axis([x1 x2 y1 y2]);
Options = odeset('RelTol', 1e-6, 'AbsTol', [1e-10 1e-10]);
dx=(x2-x1)/N; dy=(y2-y1)/N;
x=x1:dx:x2; y=y1:dy:y2;
for i=1:length(x)
for j=1:length(y)
Y0=[x(i);y(j)];
[T Y] = ode45(F, [0 tfinal], Y0, Options);
plot(Y(:,1),Y(:,2));
[T Y] = ode45(F, [0 -tfinal], Y0, Options);
plot(Y(:,1),Y(:,2));
end
end

function [] = plot_arrow(pnt,grd,xlngth,ylngth,c)

%put arrowhead on trajectory to indicate direction
%arrowhead is not scaled to indicate velocity
%so all arrowheads are the same size
%pnt is a point as a vector, [x y]
%grd is the gradient of the trajectory at pnt, e.g.[1 .5]
%c is the color of the arrowhead as a string

scaler=min(abs(xlngth),abs(ylngth)); %get length of shorter axis
if nrm>1e-6 %don't put arrows at fixed points
grd=.02*scaler*grd./nrm; %scale norm of gradient to axes
A=[0 -1;1 0]; %90 degree rotation matrix
rgrd=A*grd; %a vector perp to gradient
h=pnt+.5*grd; %the point of the arrow head
tb=pnt-.5*grd+.5*rgrd; %the top of the base of the arrow head
bb=pnt-.5*grd-.5*rgrd; %the bottom of the base of the arrow head
xs=[h(1);tb(1);bb(1)]; %x values of arrow head vertices
ys=[h(2);tb(2);bb(2)]; %y values of arrow head vertices
end
Then create the function file F. The reason you wan to name it F is because that is what it is referred to as in the phase portrait file

Code:
function yprime = F(t,y);
yprime(1) = y(1)*(3 - 2*y(1) - 1*y(2));
yprime(2) = y(2)*(5-1*y(1)-3*y(2));

yprime = yprime';
In this example, y(1) is my x and y(2) is my y. Then in the Matlab window enter in
Code:
phase_portrait(0, 1.8, 0, 2, 7, 'F', 5)
The output will be

#### Sudharaka

##### Well-known member
MHB Math Helper
Hi dwsmith,

Is this a question or something that you wish to share with others?

Kind Regards,
Sudharaka.

#### dwsmith

##### Well-known member
Hi dwsmith,

Is this a question or something that you wish to share with others?

Kind Regards,
Sudharaka.
Something to share.


text:
# PHI 103 "When I once visited Texas, it was hot. So the next time

392.9k points

"When I once visited Texas, it was hot. So the next time I visit Texas, it will be hot" is an example of a

sound inductive argument.

valid inductive argument.

sound deductive argument.

weak inductive argument.

Inductive arguments are evaluated in terms of

charm.

length.

strength.

depth.

Inductive arguments should never be characterized as

weak.

very weak.

valid.

strong.

To criticize a deductive argument logically, one might

hit the person making the argument.

show one of the premises is false.

show one of the premises is true.

show the conclusion follows validly from the premises.

Assume you are given a sound argument. What do you know about it?

It is deductive.

It is valid.

Both A and B

None of the above

In logic, arguments are never described as

true.

valid.

inductive.

sound.

One way to make an inductive argument stronger is to

ignore the argument.

offer more reasons to support the conclusion.

call the argument something in Latin.

call the argument a deductive argument.

"10 is less than 100; 100 is less than 1,000; consequently, 10 is less than 1,000" is an example of a

sound deductive argument.

valid inductive argument.

sound inductive argument.

weak inductive argument.

A valid argument is one that, if its premises are accepted as true, has

a false conclusion.

a vague conclusion.

a pointless conclusion.

a true conclusion.

If I expect that something in the future will be similar to something in the past, it is likely that I am using

deductive reasoning.

seductive reasoning.

inductive reasoning.

abductive reasoning.

389.5k points

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1. ## Series/Sequence Problem

I'm having a problem with a question

The question is: A chemical plant discharges D units of chemical molecules into a river each week. At the end of the week 90% of the chemical molecules have dissipated.

a) Find the number of units of the molecule after n weeks: I got this
an=D+.01(an-1) or an=a1+.01(an-1)

b) Estimate the amount of the molecule in the river for a very long time.
I know it involves the series of what I found in part a but how do I write that.

c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?
I don't know what do here.

Any ideas will help, Thanks

2. Originally Posted by Imuell1
I'm having a problem with a question

The question is: A chemical plant discharges D units of chemical molecules into a river each week. At the end of the week 90% of the chemical molecules have dissipated.

a) Find the number of units of the molecule after n weeks: I got this
an=D+.01(an-1) or an=a1+.01(an-1)

b) Estimate the amount of the molecule in the river for a very long time.
I know it involves the series of what I found in part a but how do I write that.

c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?
I don't know what do here.

Any ideas will help, Thanks
$A_1=D$
$A_2=D+\frac{1}{10}\left(A_1 \right) = D+\frac{1}{10}\left(D\right)= D\left(1+\frac{1}{10}\right)$
$A_3=D+\frac{1}{10}\left(A_2 \right) = D\left(1+\frac{1}{10}+\frac{1}{100}\right)$
If we continue the pattern we get
$A_n=D\sum_{i=1}^{n}\frac{1}{10^{n-1}}$

This should get you started good luck.

3. Originally Posted by TheEmptySet
$A_1=D$
$A_2=D+\frac{1}{10}\left(A_1 \right) = D+\frac{1}{10}\left(D\right)= D\left(1+\frac{1}{10}\right)$
$A_3=D+\frac{1}{10}\left(A_2 \right) = D\left(1+\frac{1}{10}+\frac{1}{100}\right)$
If we continue the pattern we get
$A_n=D\sum_{i=1}^{n}\frac{1}{10^{n-1}}$

This should get you started good luck.
Thanks.

But for part c...

Would I test for convergence with the p test or something?
or do I somehow take the limit then just set that to less than T?

4. Originally Posted by TheEmptySet
$A_1=D$
$A_2=D+\frac{1}{10}\left(A_1 \right) = D+\frac{1}{10}\left(D\right)= D\left(1+\frac{1}{10}\right)$
$A_3=D+\frac{1}{10}\left(A_2 \right) = D\left(1+\frac{1}{10}+\frac{1}{100}\right)$
If we continue the pattern we get
$A_n=D\sum_{i=1}^{n}\frac{1}{10^{n-1}}$

This should get you started good luck.
I believe it would actually be

$A_n = D\sum_{i = 1}^n\frac{1}{10^{i - 1}}$.

This is a geometric series, with $a = 1, r = \frac{1}{10}$ and $n = n$.

So $A_n = \frac{Da(1 - r^n)}{1 - r}$.

5. Originally Posted by Prove It
I believe it would actually be

$A_n = D\sum_{i = 1}^n\frac{1}{10^{i - 1}}$.

This is a geometric series, with $a = 1, r = \frac{1}{10}$ and $n = n$.

So $A_n = \frac{Da(1 - r^n)}{1 - r}$.
So I would take the limit as n approaches infinity of $\frac{Da(1 - r^n)}{1 - r}$? Which I got 1.11D.

Then for part C would I just set that limit to less than T?

I think I've got it, since the limit = 1.11D, you set that to T and solve for D? giving D=9T/10

6. Note the geometric converges for common ratio's less than 1, its the one of its most known properties. your common ration here is 1/10


text:
Procedure CaseFileSave(url, contents)

# CaseFileSave¶

The function CaseFileSave saves a specific subset of identifiers to a case file. If the file already exists, it is completely overwritten.

CaseFileSave(
url,                          ! (input) a scalar string expression
contents                      ! (input) a subset of AllIdentifiers
)

## Arguments¶

url
A string referencing the url of the case file in which you want to save the data. This url can point to a file on your local file system, or to a network location.
contents
A subset of AllIdentifiers containing all the identifiers that must be saved. Preferrably, this set is an element of AllCaseFileContentTypes such that, when reading back the case file, the content type can be determined correctly.

## Return Value¶

The procedure returns 1 on success. If any other error occurs, the procedure returns 0 and CurrentErrorMessage will contain a proper error message.

Note

• This function is only applicable if the project option Data_Management_style is set to Disk_files_and_folders.
• This function will only save the data to the specified file. It does not change the value of CurrentCase or CurrentCaseFileContentType, nor does it mark the current data as being saved.
• If your application is linked to the AIMMS PRO server, the url can also point to a case file stored at the server.
• When you save using CaseFileSave to an existing .data file with sections, the sections are removed.